Differential pricing decision-making on spot space under dynamic game of air freight transport companies

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0. 引言

In terms of game pricing, Raza et al. studied pricing issues in the competitive environment of two airlines^[21]Luo et al. studied the pricing competition problem of two airlines offering two different fare levels each^[22]Zhao et al. studied the pricing and cabin allocation decision-making problems in a duopoly market^[23]Grauberger et al. studied the problem of optimizing both the price and the number of seats for airlines in a competitive market^[24]Ko studied the pricing decision-making problem of airlines in various situations in a competitive market^[25]Wang Xiaoli et al. studied the capacity allocation and pricing issues of air cargo companies in the agency market and spot market^[26]Zhou Yinyan et al. divided the customers of air cargo companies into long-term customers and temporary customers, and assumed that long-term customers could also sell their remaining cabin space to temporary customers. Based on this, they studied the pricing problem of air cargo companies under the game theory^[27]Gao Jinmin et al. introduced the theory of "supermodel game" and studied the joint decision-making problem between airlines regarding discounted ticket pricing and cabin control^[28].

1. 问题描述

Two air cargo companies (E)₁、 E₂）When competing with each other in the spot market of a certain route, the dynamic pricing process of its cabin space is as follows: E₁、 E₂Maintain all available cabin spaces in the initial stage, and then establish cabin pricing for all sales stages. Attract booking demand for each stage through cabin pricing, and accept bookings to generate sales revenue for each stage. With the continuous sales of cabin space, E₁、 E₂The available cabin space gradually decreases, and if all cabin spaces are sold before the end of the sales period, the remaining sales stage will not be able to accept booking demands; If all seats are not sold out by the end of the final sales phase, the remaining seats will not be sold and result in waste.

Due to the short sales period in the spot market, E₁、 E₂It is difficult to adjust prices in a timely manner according to changes in market demand. In this situation, E₁、 E₂It is necessary to establish cabin pricing for all sales stages before the sales period. Assuming there is only one pricing level in each sales stage, and the pricing in each stage is from low to high, then the stagetPricing of cabin spacep_tIt will be a piecewise function.Figure 1For a cabin pricing plan consisting of 7 sales stages, the initial cabin pricing for the initial stage isaThe pricing of cabin space in the second phase has increased tobIn the fourth stage, the cabin pricing will further increase tocThe pricing of cabin space in Phase 6 has increased todThe cabin pricing for stages 3, 5, and 7 has not changed.

Figure  1.  Process of pricing decision-making

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E₁、 E₂The purpose of adopting differential pricing is to maximize sales revenue based on potential demand and the feedback characteristics of shippers on changes in cabin prices. With E₁For example, when implementing differential pricing, the following factors need to be considered: E₁、 E₂The number of available seats each holds; E₁After setting prices for each stage, E₂Possible pricing schemes; The booking requirements for each stage have an impact on E₁、 E₂Feedback relationship of pricing strategy; Revenue from cabin sales at each stage.

E₁The booking requirements for each stage depend on E₁、 E₂The demand obtained by each at the original price and the demand varying with E₁、 E₂Changes in prices. If E₁、 E₂At the stagetThe cabin pricing is as follows:p_1tandp_2tThe booking requirements are as follows:D_1tandD_2t, then in the staget(t=1, …, T）The relationship between booking demand and cabin pricing is

$\begin{array}{l}D{}_{1t}=F(p{}_{1t},p{}_{2t})(1)\\ D{}_{2t}=G(p{}_{1t},p{}_{2t})(2)\end{array}$

In the formula:F(p_1t, p_2t）For E₁Booking requirements and E₁、 E₂The functional relationship of cabin pricing;G(p_1t, p_2t）For E₂Booking requirements and E₁、 E₂The functional relationship of cabin pricing.

Due to the full transparency of the air cargo market, air cargo companies are aware of the price information of cabin seats in the market^[22]When E₁After determining the cabin pricing, E₂It is necessary to price the cabin based on equation (2), which leads to E₁The demand changes, and vice versa. Due to E₁、 E₂When pricing cabin space, it is necessary to consider the impact of the corresponding pricing strategy that the other party will adopt on their own demand, therefore, this is a game pricing problem. In addition, as the pricing of cabin seats in each stage needs to be arranged in descending order, the pricing of the previous stage determines the lower limit of the pricing of the next stage. At this point, E₁、 E₂When pricing cabin space, the goal cannot simply be to maximize profits at each stage, but to consider the constraints of pricing at each stage. Therefore, this is also a dynamic pricing problem.

2. 模型建立

2.1 Model Assumptions

The following assumptions are proposed for the dynamic game pricing model of two companies.

(1) For a flight segment that includes two air cargo companies, their service levels are the same except for cabin pricing.

(2) The demands of shippers at different stages of sales are independent of each other.

(3) The pricing of cabin space follows the order of lower prices as the sales stage begins.

(4) The demand of the two companies at each stage is related to their own pricing and the average pricing of their competitors at that stage.

(5) Not considering overbooking and cancellation of bookings.

2.2 Modeling

Based on assumption (4), the relationship between booking demand and cabin pricing at each stage of the two companies can be expressed as

$\begin{array}{l}D{}_{1t}=a{}_t-b{}_{t}p{}_{1t}+c{}_{t}p{}_{2t}(3)\\ D{}_{2t}=a{}_t-b{}_{t}p{}_{2t}+c{}_{t}p{}_{1t}(4)\end{array}$

In the formula:a_tIs a constant;b_tFor the stagetThe change in booking demand of a company when the cabin pricing changes by one unit;c_tFor the stagetThe change in booking demand of a company when the cabin pricing changes by one unit.

At any staget, must meetb_t> c_t, indicating that the booking demand of each company is more affected by its own pricing than by the pricing of competitors^[26]At this point, it can be established in E₂Under the determination of cabin pricing at each stage, E₁The cabin pricing model is

$\begin{array}{l}\pi {}_1=\max ({\displaystyle \sum _{t=1}^{{\rm T}}p}{}_{1t}q{}_{1t})(5)\\ \text{s}.\text{t}.q{}_{1t}=\min \{D{}_{1t},W{}_{1t}\}(6)\\ D{}_{1t}\ge 0(7)\\ {\displaystyle \sum _{t=1}^{{\rm T}}q}{}_{1t}\le W{}_1(8)\\ W{}_{1t+1}=W{}_{1t}-q{}_{1t}t\ne {\rm T}(9)\\ p{}_{1t}\le p{}_{1t+1}t\ne {\rm T}(10)\end{array}$

In the formula:π₁For E₁The maximum profit for all sales stages;W₁For E₁The initial available space for sale;W_1tandq_1tDivided into stagestWhen E₁Remaining available cabin space and accepted booking capacity.

The objective function equation (5) represents E₁Maximizing the total revenue from cabin sales at all stages; Equation (6) represents E₁The booking volume accepted at each stage cannot exceed its booking demand for that stage, nor can it exceed its remaining cabin capacity for that stage; Equation (7) represents E₁The booking demand at each stage is non negative; Equation (8) represents E₁The sum of bookings accepted at all stages cannot exceed the initial available space for sale; Equation (9) represents E₁The conversion relationship of the remaining available cabin space in each stage; Equation (10) represents E₁Pricing the cabin in the order of lower prices as the sales stage begins.

Similarly, when E₁When determining the pricing of cabin space at each stage, E₂The cabin pricing model is

$\begin{array}{l}\pi {}_2=\max ({\displaystyle \sum _{t=1}^{{\rm T}}p}{}_{2t}q{}_{2t})(11)\\ \text{s}.\text{t}.q{}_{2t}=\min \{D{}_{2t},W{}_{2t}\}(12)\\ D{}_{2t}\ge 0(13)\\ {\displaystyle \sum _{t=1}^{{\rm T}}q}{}_{2t}\le W{}_2(14)\\ W{}_{2t+1}=W{}_{2t}-q{}_{2t}t\ne {\rm T}(15)\\ p{}_{2t}\le p{}_{2t+1}t\ne {\rm T}(16)\end{array}$

In the formula:π₂For E₂The maximum profit for all sales stages;W₂For E₂The initial available space for sale;W_2tandq_2tDivided into stagestWhen E₂Remaining available cabin space and accepted booking capacity.

3. Solution method

Based on the model presented in this article, two problems need to be solved: the dynamic pricing problem of another company when the cabin pricing of one company is determined; The pricing problem of game theory when two companies compete with each other. For dynamic pricing problems, they can be transformed into convex quadratic programming problems and solved accordingly; For game pricing problems, iterative methods are used to solve them^[29]The specific process is as follows.

3.1 Solving dynamic pricing problems

With E₂When determining cabin pricing, E₁Taking dynamic pricing as an example, E can be used at this point₁The relationship function between booking demand and cabin pricing is expressed as

$\begin{array}{l}D{}_{1t}=a^{\prime }{}_t-b{}_{t}p{}_{1t}(17)\\ a^{\prime }{}_t=a{}_t+c{}_{t}p{}_{2t}(18)\end{array}$

At the stagetIfD_1t> W_1tSo E₁As long as the cabin pricing is increased toD_1t=W_1tThe degree to which E can increase profits indicates that E₁The pricing strategy has not reached its optimal level, indicating that under the optimal pricing strategy, at any stagetshall beD_1t≤W_1tThus, equation (6) can be adjusted to

$q{}_{1t}=D{}_{1t}(19)$

According to equations (17) to (19), when E₂When determining the pricing of cabin space at each stage, E₁The cabin pricing model can be transformed into

3.2 博弈定价问题求解

4. 实证分析

4.1 实例数据

Figure  2.  Booking ratios of passengers at each period at Dalian-Guangzhou segment

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Figure  3.  Freight volumes of two companies at each period

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Table  1.  Function parameters

阶段	1	2	3	4	5	6	7	8	9	10	11
at	369.3	483.9	305.6	673.4	382.4	445.7	433.3	803.3	498.1	1 107.0	624.7
bt	21.7	28.4	17.9	58.1	17.8	26.1	20.8	33.2	10.6	74.2	27.4
ct	10.8	14.2	9.0	29.0	8.9	13.1	10.4	16.6	5.3	37.1	13.7

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4.2 计算结果

Figure  4.  Space price at each period under differential pricing mode

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Figure  5.  Revenues at each period of airport freight transport under different pricing modes

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4.3 敏感性分析

Figure  6.  Pricing strategies of two companies under different values of k

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Figure  7.  Revenues of two companies under different values of k

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5. 结语