A Design & Optimization Method for Intercity Demand-responsive Transit Considering Passengers' Choice Behavior
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摘要: 需求响应式公交是1种基于预约的新型出行服务模式。然而，现有研究主要关注城市内部场景，而在城际场景下的服务设计与优化仍显不足。同时，现有模型通常侧重于以运营成本最低或运营利润最大为目标，缺乏对运营利润与消费者剩余的综合考量，难以全面提升社会福利。此外，现有研究普遍假设固定需求密度，未充分纳入乘客选择偏好。为弥补以上不足，在综合考虑乘客的选择行为的基础上，以社会福利最大化(运营利润与消费者剩余之和)为目标，选取发车间隔、主城与卫星城的服务区域面积比、票价作为决策变量，基于连续近似法构建城际需求响应式公交服务设计优化模型，并采用拉格朗日乘子法进行求解。进一步对比社会福利最大化与运营利润最大化2种目标下的最优方案。结果表明：在社会福利最大化目标下，虽运营利润为负，但在适度补贴条件下，社会福利可由54.976元/h提升至110.906元/h，验证了方法的有效性。对模型部分参数的灵敏度分析结果表明：卫星城服务区域面积对社会福利影响较小，并非城际需求响应式公交布局的主要限制因素；而城际公路长度对乘客选择影响显著，需通过调查确保际需求响应式公交合理布局。总体而言，本文提出的城际需求响应式公交服务设计优化方法，不仅能够为决策者在给定需求密度条件下优化发车间隔、服务区域面积与票价提供参考，还可结合灵敏度分析结果，辅助其选择合适车型，以进一步提升整体运营效益与社会福利。Abstract: Demand-responsive transit is an emerging reservation-based travel service. However, existing studies have mainly focused on intracity context, while studies on service design and optimization in intercity context re-main limited. Meanwhile, current models typically pursue a single operator-oriented objective, such as operational cost minimization or profit maximization, but rarely incorporate operation profits and consumer surplus, thereby lim-iting their ability to enhance overall social welfare. Additionally, most studies assume fixed demand density and fail to adequately involve passengers'choice preferences. To address these shortcomings, this study incorporates passen-gers'choice behavior and develops a design & optimization model for intercity demand-responsive transit (IDRT) service. The objective of this model is maximizing social welfare, which is defined as the sum of operation profits and consumer surplus. The decision variables include departure interval, service-area ratio between the major city and satellite cities, and ticket fare. The model is formulated using the continuous approximation method and solved with the Lagrange multiplier method. Furthermore, the optimal solutions of models with social welfare maximiza-tion and operation profit maximization are compared. Results show that although operation profit is negative in the model of social welfare maximization, an appropriate subsidy can raise social welfare from 54.976 CNY/h to 110.906 CNY/h, demonstrating the effectiveness of the proposed method. The results of the sensitivity analysis on se-lected model parameters indicate that the service area of satellite cities has only a minor impact on social welfare and is not a key constraint in IDRT layout, whereas length of intercity highway has a significant influence on passen-gers'choice behavior, highlighting the importance of empirical surveys for rational deployment of IDRT layout. Over-all, the proposed method not only provides decision-makers with a practical tool to determine departure intervals, ser-vice-area allocation, and ticket fares given demand density conditions, but also assists them in selecting appropriate vehicle types based on sensitivity analysis, thereby improving both operational efficiency and social welfare.
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图 2 卫星城服务区域潜在需求密度对模型结果的影响
Figure 2. Impacts of potential demand density in satellite city on the model results
图 3 主城服务区域潜在需求密度对模型结果的影响
Figure 3. Impacts of potential demand density in main city on key the model results
图 4 卫星城服务区域面积对模型结果的影响
Figure 4. Impacts of service area of satellite city on key the model results
表 1 模型符号说明
Table 1. Notation of symbols
符号 含义 单位 基准值 A1 卫星城服务区域面积 km2 5 A2 主城服务区域面积 km2 a 主城服务区域与卫星城服务区域面积之比(= A2/A1) L 城际公路长度 km 60 N 车队规模 辆 S 车座数 座/辆 30 H 车辆运行周期时间期望 h h 发车间隔 h f 票价(单程票价) 元 b1 车辆运营固定成本系数 元/(辆· h) 30 b2 车辆运营可变成本系数 元/(座· h) 0.3 c 单位车辆运营成本(= b1 + b2 · S) 元/(辆· h) 30 + 0.3 ×30 C 单位时间运营成本 元/h D1 车辆从卫星城服务区到主城服务区的距离期望 km D2 车辆从主城服务区到卫星城服务区的距离期望 km D 车辆运行1个周期的距离期望 km n1 卫星城服务区车辆单次服务乘客数量 人 n2 主城服务区车辆单次服务乘客数量 人 k 旅行商问题最短路近似公式系数 1.15 v1 卫星城内平均车速 km/h 25 v2 主城内平均车速 km/h 20 vL 城际公路平均车速 km/h 60 Q1 卫星城的实际需求密度 人/(km2· h) Q2 主城的实际需求密度 人/(km2· h) ew 等待时间弹性系数 0.35 ev 车内时间弹性系数 0.175 ep 票价弹性系数 0.07 z 等待时间近似调整系数 0.5 q1 卫星城服务区的乘客潜在需求密度 人/(km2· h) 10 q2 主城服务区的乘客潜在需求密度 人/(km2· h) 40 $\bar{H}_{1}$ 卫星城乘客平均车内时间 h $\bar{H}_{2}$ 主城乘客平均车内时间 h W 单位时间社会福利 元/h P 单位时间运营利润 元/h R 单位时间运营收入 元/h G 单位时间总消费者剩余 元/h G1 卫星城服务区域内单位时间消费者剩余 元/h G2 主城服务区域内单位时间消费者剩余 元/h J1 卫星城消费者在单个发车间隔内的票价总支出 元 J2 主城消费者在单个发车间隔内的票价总支出 元 U1 卫星城服务区域内单位面积单位时间消费者收益 元/(km2· h) U2 主城服务区域内单位面积单位时间消费者收益 元/(km2· h) 
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表 2 基于基准值的结果
Table 2. Baseline result
目标函数 决策变量 目标函数相关变量 a h f W P G1 G2 社会福利最大化 1.027 0.467 2.774 110.906 -97.624 65.315 143.216 经营者利润最大化 0.991 0.841 4.716 54.976 8.029 18.752 28.195
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