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Reinforcement learning framework for freight demand forecasting to support operational planning decisions

Lama Al Hajj Hassan, Hani S. Mahmassani*, Ying ChenTransportation Center, Northwestern University, Chambers Hall, 600 Foster Street, Evanston, IL 60208, United States
西北大学交通中心, 钱伯斯大厅, 600 Foster Street, Evanston, IL 60208, United States


Keywords: 关键字:

Freight demand forecasting
Time series 时间序列
Reinforcement learning 强化学习
Rolling horizon 滚动地平线

Abstract 抽象

A B S T R A C T Freight forecasting is essential for managing, planning operating and optimizing the use of resources. Multiple market factors contribute to the highly variable nature of freight flows, which calls for adaptive and responsive forecasting models. This paper presents a demand forecasting methodology that supports freight operation planning over short to long term horizons. The method combines time series models and machine learning algorithms in a Reinforcement Learning framework applied over a rolling horizon. The objective is to develop an efficient method that reduces the prediction error by taking full advantage of the traditional time series models and machine learning models. In a case study applied to container shipment data for a US intermodal company, the approach succeeded in reducing the forecast error margin. It also allowed predictions to closely follow recent trends and fluctuations in the market while minimizing the need for user intervention. The results indicate that the proposed approach is an effective method to predict freight demand. In addition to clustering and Reinforcement Learning, a method for converting monthly forecasts to long-term weekly forecasts was developed and tested. The results suggest that these monthly-to-weekly long-term forecasts outperform the direct long term forecasts generated through typical time series approaches.
A B S T R A C T 货运预测对于管理、规划、运营和优化资源使用至关重要。多种市场因素导致货运流量的高度可变性,这需要适应性和响应性预测模型。本文提出了一种需求预测方法,该方法支持短期到长期的货运运营计划。该方法将时间序列模型和机器学习算法结合在强化学习框架中,应用于滚动地平线。目标是开发一种有效的方法,通过充分利用传统的时间序列模型和机器学习模型来减少预测误差。在一家美国多式联运公司集装箱运输数据的案例研究中,该方法成功地降低了预测误差幅度。它还允许预测密切关注市场的最新趋势和波动,同时最大限度地减少对用户干预的需求。结果表明,所提方法是一种有效的货运需求预测方法。除了聚类和强化学习之外,还开发并测试了一种将月度预测转换为长期周度预测的方法。结果表明,这些每月到每周的长期预测优于通过典型时间序列方法生成的直接长期预测。

1. Introduction 1. 引言

Taking 2015 as the base year, total U.S. freight shipments are expected to increase 41 percent by 2045 (Assoc. of Am. Railroads, 2017). In the midst of this growing demand, intermodal operations, which combine truck drayage with rail long haul service, offer a viable option to serve various markets. These operations help reduce trucks on highways at a competitive price. Intermodal traffic is expected to grow as the trucking industry continues to face driver shortages and regulatory changes that raise costs and limit driver hours and miles. The projected cost increases in trucking coupled with growth in e-commerce will further push retailers to seek savings along the supply chain (Stephens, 2017).
To accommodate the projected growth, intermodal carriers need to understand and properly react to regulations, market forces, and evolving customer requirements in a competitive market. The response should be reliable through the coordination between freight and equipment flow (Dewitt and Clinger, 2000). Customers and retailers are expecting more shipping options and higher delivery speeds and reliability, lower prices, flexible destinations, and distribution terminals closer to urban agglomerations (Choe et al., 2017).
为了适应预期的增长,多式联运承运人需要了解竞争激烈的市场中的法规、市场力量和不断变化的客户需求并做出适当的反应。通过货运和设备流之间的协调,响应应该是可靠的(Dewitt和Clinger,2000年)。客户和零售商期望有更多的运输选择、更高的交付速度和可靠性、更低的价格、灵活的目的地以及更靠近城市群的配送终端(Choe et al., 2017)。
These challenges underscore the role of data in planning and managing the supply chain process. Regardless of the mode in question, operators need forecasting models that not only analyze and project based on historical trends and market information, but also adapt to recent changes in customer purchase patterns, policy and so on. Extracting analytical insights from the network data, developing demand projections on a real-time basis, and acting on them will affect carriers' market position (Choe et al., 2017). This dynamic insight-response mechanism promises improved fleet utilization and business growth.
这些挑战凸显了数据在规划和管理供应链流程中的作用。无论采用何种模式,运营商都需要的预测模型不仅要根据历史趋势和市场信息进行分析和预测,还要适应客户购买模式、政策等的最新变化。从网络数据中提取分析见解,实时开发需求预测并采取行动将影响运营商的市场地位(Choe et al., 2017)。这种动态的洞察-响应机制有望提高车队利用率和业务增长。
Freight demand has been modeled using regression and time series techniques, behavioral models (Regan & Garrido, 2001), commodity-based input-output models, inventory theory, truck route optimization and simulation approaches (Mahmassani et al., 2007). One critical limitation in existing freight demand models is the inability to adapt quickly to new conditions (Chow et al., 2010). In addition, some models require extensive demand and economic datasets making implementation time and resource consuming (Cambridge Systematics, 2010). This paper presents a demand forecasting methodology to support operational planning over short to long term horizons. The approach combines time series models ("forecasters") in a Reinforcement Learning (RL) framework implemented over a rolling horizon. In the proposed methodology, predictions for each market are generated after clustering the market lanes (different origin-destination pairs) on the basis of observed container demand patterns. The contribution of this paper lies in developing a structured responsive approach that starts by identifying market clusters in a large dataset, and then establishing a suitable subset (committee) of forecasters for each cluster. Constructing a committee of forecasters instead of relying on a single predictor helps improve the prediction accuracy since no one model is always the best performer (Newbold and Granger, 1974; Granger and Newbold, 1976; Granger and Jeon, 2004; Yang, 2004; Bichpuriya et al., 2016). The approach uses RL logic to overcome the responsiveness limitation of time series models by learning from the recent performance of the committee members. Reinforcement Learning is a computational approach to learning in which the agent learns and adapts through continuous experimentation. An agent's goal is to maximize a suitably defined reward function, so the agent explores several options and consequently discovers which action yields the highest return, and as a result learns what to do (Sutton and Barto, 1998).
货运需求已使用回归和时间序列技术,行为模型(Regan&Garrido,2001),基于商品的投入产出模型,库存理论,卡车路线优化和模拟方法(Mahmassani等人,2007)进行建模。现有货运需求模型的一个关键限制是无法快速适应新条件(Chow et al., 2010)。此外,一些模型需要大量的需求和经济数据集,这使得实施耗时耗资(Cambridge Systematics,2010)。本文提出了一种需求预测方法,以支持短期到长期的运营规划。该方法将时间序列模型(“预测器”)结合在强化学习 (RL) 框架中,该框架在滚动范围内实施。在所提出的方法中,根据观察到的集装箱需求模式,在对市场航线(不同的始发地-目的地对)进行聚类后,生成对每个市场的预测。本文的贡献在于开发一种结构化的响应方法,该方法首先在大型数据集中识别市场集群,然后为每个集群建立一个合适的预测者子集(委员会)。建立一个预测者委员会而不是依赖单一的预测器有助于提高预测准确性,因为没有一个模型总是表现最好的(Newbold and Granger, 1974;Granger 和 Newbold,1976 年;Granger 和 Jeon,2004 年;杨, 2004;Bichpuriya 等人,2016 年)。该方法使用 RL 逻辑,通过学习委员会成员最近的表现来克服时间序列模型的响应性限制。强化学习是一种计算学习方法,其中智能体通过不断的实验来学习和适应。 智能体的目标是最大化适当定义的奖励函数,因此智能体探索了几个选项,从而发现哪个行为产生了最高的回报,并因此学会了该做什么(Sutton和Barto,1998)。
In the following section, a review of the relevant literature is presented, followed by a presentation of the methodology in Section 3. The model is then tested, and the results are discussed in Section 4. Finally, Section 5 concludes the paper and suggests future extensions of this work.
在下一节中,将对相关文献进行回顾,然后在第3节中介绍方法。然后对模型进行测试,结果将在第 4 节中讨论。最后,第5节总结了本文,并提出了这项工作的未来扩展。

2. Literature review 2. 文献综述

2.1. Freight demand modelling
2.1. 货运需求建模

Several tools have been developed for freight demand modeling to support public and private sector decision making processes (Chow et al., 2010; Tavasszy and De Jong, 2013). Freight demand has been modeled using regression and time series techniques (Fite et al., 2002; Farrington and Harris, 2011), behavioral models (Regan and Garrido, 2001; Ben-Akiva et al., 2008), spatial econometrics techniques (Garrido and Mahmassani, 1998, 2000), commodity-based input-output models (Black, 1999; Holguín-Veras and Patil, 2008; Cascetta et al., 2013), inventory and supply chain theories (Winston, 1983), truck route optimization and simulation approaches (Zhang et al., 2008; Cambridge Systematics, 2010; Nuzzolo et al., 2013).
已经开发了几种用于货运需求建模的工具,以支持公共和私营部门的决策过程(Chow et al., 2010;Tavasszy 和 De Jong,2013 年)。货运需求已使用回归和时间序列技术进行建模(Fite et al., 2002;Farrington和Harris,2011),行为模型(Regan和Garrido,2001;Ben-Akiva et al., 2008)、空间计量经济学技术(Garrido and Mahmassani, 1998, 2000)、基于商品的投入产出模型(Black, 1999;Holguín-Veras 和 Patil,2008 年;Cascetta et al., 2013)、库存和供应链理论(Winston, 1983)、卡车路线优化和模拟方法(Zhang et al., 2008;剑桥系统学,2010 年;Nuzzolo等人,2013)。
Behavioral models simulate shippers' decision processes and carrier choices. These can be implemented if survey data are collected from shippers and carriers; however the approach can be time-consuming and not readily available as shipper-carrier interaction data are typically proprietary (Cambridge Systematics, 2010). Input-output models are used to determine the amount of freight by commodity consumed and produced in a region as a result of economic mechanisms. Truck trips are then estimated based on the calculated freight flows. To develop such models, one would require historical and predicted freight flow data and economic indicators. Inventory models simulate the decision-making process between customers and suppliers while seeking to minimize the total transportation cost (shipping, inventory and ordering). Simulation and network models focus on freight mode choice and routing decisions (Mahmassani et al., 2007; Cambridge Systematics, 2010). Time series models are commonly used by public sector agencies and have fewer resource requirements compared to other approaches. However, time series models require a long history of observations and assume previous conditions remain stationary into the future. Therefore, on their own such models do not adapt to new situations and information quickly.
行为模型模拟托运人的决策过程和承运人选择。如果从托运人和承运人那里收集调查数据,则可以实施这些措施;然而,这种方法可能很耗时,而且不容易获得,因为托运人与承运人的互动数据通常是专有的(Cambridge Systematics,2010)。投入产出模型用于确定由于经济机制而在一个地区消费和生产的商品的货运量。然后根据计算出的货运流量估算卡车行程。要开发这样的模型,需要历史和预测的货运流量数据和经济指标。库存模型模拟客户和供应商之间的决策过程,同时寻求将总运输成本(运输、库存和订购)降至最低。仿真和网络模型侧重于货运模式选择和路线决策(Mahmassani 等人,2007 年;剑桥系统学,2010 年)。时间序列模型通常由公共部门机构使用,与其他方法相比,它需要的资源更少。然而,时间序列模型需要很长的观测历史,并假设以前的条件在未来保持静止。因此,这些模型本身并不能快速适应新情况和信息。
One of the critical issues facing existing tools include lack of and late responsiveness to consumer demand variations and to sudden economic conditions (Cambridge Systematics, 2010). This misalignment between the freight market and the model adjustments could result in missed opportunities and inefficient operations. As a result, this paper uses time-series and machine learning models within a reinforcement learning approach to leverage available data and fast responsiveness to new conditions. Freight demand modeling literature using time series analysis have relied on single models to generate point forecasts (Farrington and Harris, 2011) or multiple models to generate bounds (Liu et al., 2006). The contribution of this paper lies in developing and testing using actual carrier data a structured responsive approach that starts by identifying clusters in the data and their suitable committee of forecasters. The approach then uses the RL logic to overcome the responsiveness limitation of time series models and learn from the recent performance of the committee members (model components).
现有工具面临的关键问题之一是对消费者需求变化和突如其来的经济状况缺乏反应和反应迟钝(Cambridge Systematics,2010)。货运市场与模型调整之间的这种错位可能导致错失机会和运营效率低下。因此,本文在强化学习方法中使用时间序列和机器学习模型来利用可用数据和对新条件的快速响应。使用时间序列分析的货运需求建模文献依赖于单个模型来生成点预测(Farrington and Harris,2011)或多个模型来生成边界(Liu et al.,2006)。本文的贡献在于使用实际的载波数据开发和测试一种结构化的响应方法,该方法首先识别数据中的聚类及其合适的预报员委员会。然后,该方法使用 RL 逻辑来克服时间序列模型的响应性限制,并从委员会成员(模型组件)的近期表现中学习。

2.2. Reinforcement learning
2.2. 强化学习

Inspired by behavioral psychology, Reinforcement Learning has been adopted and adapted by multiple disciplines such as game theory, control theory, and operations research, as a mechanism for integrating information from various sources and/or outcomes from repeated trials. In reinforcement learning, given a set of possible actions, the agent discovers the most rewarding decision by trying an action and reaping the reward immediately or after a set of subsequent actions i.e., delayed reward. Because of the structure

of this problem, the agent is required to make a tradeoff between exploration of new options that may have higher rewards and exploitation of past actions that led to rewards (Weigang et al., 2008).
对于这个问题,智能体需要在探索可能具有更高奖励的新选项和利用导致奖励的过去行为之间做出权衡(Weigang et al., 2008)。
In the field of game theory, Erev and Roth (1998) studied the descriptive (best fit) and predictive performance of reinforcement learning models in games with a unique equilibrium. They showed that static equilibrium predictions were outperformed by those from reinforcement learning models. They tested three games with similar equilibrium conditions and noted that the speed at which each experimental subject converged to the equilibrium prediction varied depending on their starting point. They further tested the long-term ramifications of initial conditions on the learning process and argued that although it is possible to improve the initial estimates their effects on the model were limited.
In earlier work, Roth and Erev (1995) introduced two psychological features in their reinforcement learning framework: experimentation and recency effects. By allowing experimentation in the model, a subject's choice set was not limited to past successful ones, thereby preventing subjects from being rapidly locked into one sub-optimal choice. Recency effects recognize that recent events might have a more significant impact on decisions compared to past experience. These two features allow the reinforcement learning model to be more responsive to changes in the environment, especially under partial and general conditions of information availability.
Feltovich (2000) designed a multi-stage game and compared Nash equilibrium predictions to those of reinforcement learning and belief based models. In reinforcement learning models, a player bases his/her choices only on payoffs of past actions whereas in belief based models choices depend on the expected payoff of an action given held beliefs of the likely actions of the other players. Using a set of pre-defined criteria, the author concluded that predictions generated from both learning models substantially outperform those of Nash equilibrium. In fact, in the tested games and after multiple repetitions, the predictions of both models started and remained far from equilibrium. The experimental results were inconclusive as to which learning model (Reinforcement or Belief-Based) outperformed the other, though both models resulted in similar behavioral patterns regardless of their different specifications, suggesting that there might not be a single perfect model that would capture behavior under all circumstances.
While there are numerous examples in the literature for the use of machine learning techniques for planning and management of airport operations, forecasting models for freight operations have been more inclined toward the use of time series and multiple regression models (Bodily and Freeland, 1988; Guerrero and Elizondo, 1997; Fite et al., 2002; Patil and Sahu, 2016). An overview of the use of reinforcement learning for forecasting demand for transportation services and the models used in forecasting freight flow are presented in Table 1.
虽然文献中有许多使用机器学习技术规划和管理机场运营的例子,但货运运营的预测模型更倾向于使用时间序列和多元回归模型(Bodily and Freeland,1988;格雷罗和埃利桑多,1997年;Fite 等人,2002 年;Patil 和 Sahu,2016 年)。表1概述了使用强化学习预测运输服务需求以及用于预测货运流量的模型。
Table 1 shows that reinforcement learning succeeded in guiding processes and improving decisions in the aviation sector. Review of freight-related literature revealed that there had been attempts to incorporate several machine learning techniques in the forecasting process although time series models are still adopted. This paper seeks to combine the already established time series models with reinforcement learning to advance freight forecasting models and their application to support operational planning decision processes.

3. Methodology 3. 方法论

This section presents the overall approach to forecasting freight movement. Assume that we have oracles or forecasters who provide a forecast for each time period in a given market-in this case, the number of containers to be moved in a given week in that market. No forecast is perfect, and sometimes a forecast will hit the mark or get very close, whereas at other times it may miss the mark by varying degrees. Not knowing the past performance of the oracles, the operator (using the forecasts as a basis for planning decisions) may take a simple average of the forecasts as the "consensus" forecast. However, recognizing that some forecasters outperform others in certain instances or for different periods of time, the operator will place different weights on the respective forecasts in forming the "final" forecast. Reinforcement Learning (RL) provides a mechanism for (1) dynamically learning about the respective performance of the different forecasters based on the quality of their past forecasts, placing greater emphasis on more recent instances, and (2) weighing the respective forecasts accordingly at each instance.
本节介绍预测货运的总体方法。假设我们有 预言机或预测者,他们为给定市场中的每个时间段提供预测 - 在本例中,该市场在给定的一周内要移动的集装箱数量。没有一个预测是完美的,有时预测会达到目标或非常接近,而在其他时候,它可能会不同程度地偏离目标。由于不知道预言机的过去表现,操作员(使用预测作为计划决策的基础)可能会将预测的简单平均值作为“共识”预测。然而,考虑到某些预测者在某些情况下或不同时间段的表现优于其他预测者,运营商将在形成“最终”预测时对各自的预测施加不同的权重。强化学习 (RL) 提供了一种机制,用于 (1) 根据不同预测者过去预测的质量动态学习各自的表现,更加强调最近的实例,以及 (2) 在每个实例中相应地权衡各自的预测。
In this context, the forecasters, or agents, consist of different statistical or Machine Learning models developed using the same training (estimation) data set, and implemented in a rolling horizon framework to provide forecasts at different time scales. These are run in parallel, and compared against actual realizations once those have materialized, providing an automated basis for scoring the forecasters' respective performance for use in the RL weight updating mechanism. The challenge consists in specifying the reward function and formulating the updating mechanism. The mathematical details are provided in this section as part of the model formulation.
在这种情况下,预测者或代理由使用相同的训练(估计)数据集开发的不同统计或机器学习模型组成,并在滚动地平线框架中实现,以提供不同时间尺度的预测。这些是并行运行的,并在实现后与实际实现进行比较,为对预测者各自的表现进行评分提供自动化基础,以用于 RL 权重更新机制。挑战在于指定奖励函数和制定更新机制。本节将作为模型公式的一部分提供数学详细信息。
Before developing and applying the forecasting models, container demand originating in a given market is clustered into destination lane groups with similar spatio-temporal patterns, as described in Section 3.1. The overall approach is summarized in Fig. 1. The formulation of the continuous, self-correcting RL model is presented in Section 3.2, followed in Section 3.3 by a description of the time scales over which the forecasting models are applied. Recognizing the presence of seasonality and special events (e.g. holidays), Section 3.4 presents the approach followed to incorporate these effects in the models. Finally Section 3.5 describes the metrics used to evaluate the performance of the model in the case study application.
在开发和应用预测模型之前,将源自给定市场的集装箱需求聚类到具有相似时空模式的目的地航线组中,如第 3.1 节所述。图 1 总结了总体方法。第 3.2 节介绍了连续、自校正 RL 模型的公式,然后在第 3.3 节中描述了应用预测模型的时间尺度。认识到季节性和特殊事件(例如节假日)的存在,第3.4节介绍了将这些影响纳入模型所采用的方法。最后,第 3.5 节描述了用于评估案例研究应用程序中模型性能的指标。

3.1. Market lane clustering
3.1. 市场通道聚类

After correcting for missing data and data errors, the primary markets are identified along with their trends in time and space. The primary markets are defined as the origin terminals where all the containers are loaded. The primary market is the one for which
Table 1 表1
Selected freight forecasting methods and reinforcement learning applications.
Reference Method Application Findings
(GOSAVII et al., 2002)
(GOSAVII 等人,2002 年)
Formulated the problem as a semi-Markov
decision problem aiming to maximize the
average reward. 平均奖励。
Objective values defined for each state-action
pair were updated within a neural network
Revenue management problem for
a single flight leg
The proposed method outperforms the
Expected Marginal Seat Revenue (EMSR),
预期边际座位收入 (EMSR),
a heuristic that is widely used in the
Weigang et al., 2008
Weigang 等人,2008
Simulate future airspace demand to identify
capacity requirements in each sector for
different periods of time.
Decision support process is designed as a
Markov Decision Chain (MDC), the state
马尔可夫决策链 (MDC),状态
information from MDC is transferred to a
来自 MDC 的信息被传输到
reinforcement learning model in which an
action is selected, executed, and its
corresponding outcomes are used to update
the learning process. 学习过程。
Air traffic flow management
Using traffic flow information for Brazil,
the model-suggested course of action was
close to reality and even led to
improvement in certain instances.
Tumer and Agogino (Tumer
Tumer 和 Agogino (Tumer
and Agogino, 2009) 和 Agogino,2009 年)
Multi-agent model that responded quickly to
weather and airport conditions to limit the
local delays. 本地延误。
These agents learn continuously through
reinforcement learning and provide air traffic
controllers with recommendations and
Air traffic management systems
A simulation based on US airspace
showed that the proposed model could
improve ATM while retaining current
flow management procedures, without
significant policy shifts. Comparison of
simulation results to outputs of a Monte
Monte 输出的仿真结果
Carlo estimation procedure revealed that
the adaptive agents outperformed the
reference case. 参考案例。
Garrido and Mahmassani 加里多和马赫马萨尼
(Garrido and
Mahmassani, 2000) Mahmassani,2000年)
Multinomial probit (MNP) model for freight
货运的多项式概率 (MNP) 模型
demand analysis and flow distribution
prediction that captures general spatial and
temporal correlation patterns.
Given order patterns and information
regarding socioeconomic activity, the model
forecasts freight flow over space and time for
operational and tactical level planning
Motor carrier company dataset of
all shipments picked up in the
state of Texas between June 1994
and July 1995 .
和 1995 年 7 月。
While predicted probabilities differed
from those in the forecasting sample, the
modified probit model succeeded in
ranking and identifying sites with a
higher probability of generating
shipments at a given time.
Moscoso-López et al. 莫斯科-洛佩斯等人。
(Moscoso-López et al., (Moscoso-López等人,
Compared the performance of Artificial
Neural Networks (ANN) and Support Vector
神经网络 (ANN) 和支持向量
Machines (SVMs) models in predicting freight
预测货运的机器 (SVM) 模型
Fresh vegetable transportation
through RO-RO operations in the
Port of Algeciras Bay
港口 Algeciras Bay
The SVMs models performed slightly
SVM 模型的表现略有
better than ANN in forecasting the volume
在预测数量方面优于 ANN,
of fresh vegetables moved on each day
predictions are generated. Primary vs secondary notation is used only to differentiate between the origin market and the destination markets respectively. It is important to examine the trends since the intensity and volatility of the fluctuations in each market hint to the type of the time series models that would best capture the different characteristics of demand such as seasonality and periodicity.
For each market, the corresponding demand dataset is clustered so as to identify groups of destination lanes. Each group consists of lanes with similar patterns and for which separate cluster-customized models are developed. Cluster analysis seeks to find groups in a dataset that increase homogeneity in a group while increasing heterogeneity between groups (Kaufman and Rousseeuw, 2009). There are various algorithms for clustering; here we employ k-means clustering approach. K-means clustering is an unsupervised approach that starts by locating centroids in the dataset and then assigning elements in the data to that centroid to create groups. The assignment seeks to minimize the sum of the squared distance between the elements and the centroid in a cluster (Schoier and Gregorio, 2017). Orlin et al. (2017) used clustering to reduce errors in sales forecasting for a retail company. Compared to the nonclustered approach, clustering the different items sold by the retail company lowered the average errors by at least 10% (Orlin et al., 2017). Hence, in this study, a market is broken down into various clusters, as described in the case study application in Section 4. For each cluster, a set of forecasting models are selected to comprise the components that are later used in the RL approach to generate the final forecast. Therefore, the predicted demand for a cluster is the weighted average of the predictions of the individual component models and the weights are updated continuously using RL.
对于每个市场,对相应的需求数据集进行聚类,以识别目的地车道组。每个组都由具有相似模式的车道组成,并为其开发单独的集群定制模型。聚类分析试图在数据集中找到增加组同质性的组,同时增加组之间的异质性(Kaufman和Rousseeuw,2009)。有多种聚类算法;在这里,我们采用k-means聚类方法。K-means 聚类是一种无监督方法,它首先在数据集中定位质心,然后将数据中的元素分配给该质心以创建组。该任务旨在最小化集群中元素和质心之间的平方距离之和(Schoier和Gregorio,2017)。Orlin等人(2017)使用聚类来减少零售公司销售预测中的错误。与非聚类方法相比,对零售公司销售的不同商品进行聚类可将平均误差降低至少 10%(Orlin 等人,2017 年)。因此,在本研究中,市场被分解为不同的集群,如第 4 节的案例研究应用程序中所述。对于每个聚类,选择一组预测模型来包含稍后在 RL 方法中使用以生成最终预测的组件。因此,对集群的预测需求是各个组件模型预测的加权平均值,并且使用 RL 不断更新权重。

3.2. Continuous and self-correcting model formulation
3.2. 连续和自校正模型的制定

In developing the forecasting model, the longitudinal dataset is divided into two time periods. The first is used for training, and the later one is used for testing. The estimation set is used for calibrating the statistical time series models and training the machine learning algorithms; forecasts are then produced using these models for the testing portion, along with forecasting new time intervals, where each time interval is one week in this application.
The demand forecasting process is designed to operate on a rolling horizon basis, whereby a forecasting horizon is updated (rolled) at the end of every week as new information becomes available. Adopting this rolling horizon approach, depicted in Fig. 2,
需求预测过程旨在以滚动为基础运行,即在每周结束时随着新信息的出现而更新(滚动)。采用这种滚动地平线方法,如图 2 所示,
Fig. 1. Stages in the presented methodology.
图 1.所介绍的方法的阶段。
allows updating the forecasts based on the most recent information. It is accomplished by adding to the estimation dataset the number of moves that occurred in the latest week, dropping the earliest week, and then forecasting. As a result, forecasting becomes a quasi-continuous process that is not only performed at the end of the forecasting horizon.
Deploying a continuous model may be directly correlated with the need to increase monitoring efforts and incurring operational inefficiencies. However, the suggested framework incorporates reinforcement learning to maximize efficiency through better deployment and informed execution. Reinforcement learning (RL) would allow the model to adapt to recent trends while eliminating the need for direct auditing. The RL notion is introduced in a manner that guides forecasts generated in this proposed approach.
Since no time series model is capable of fully capturing the demand trend characteristics and one model may not be suitable for
Fig. 2. Rolling horizon implementation.
图 2.滚动地平线实现。
forecasting at all times, this proposed approach uses multiple time series models as RL agents. The time series or machine learning models (agents) are chosen based on the initial trend analysis performed at the level of clusters. The forecast of the individual component (agent) is then weighed through the reinforcement learning mechanism to obtain the final forecast. Suppose time series models, , are used for forecasting, where is the -th time step, denote the prediction accuracy of component , which is the fraction of time over the testing period in which component gives a correct prediction.
在任何时候进行预测,所提出的方法使用多个时间序列模型作为RL代理。时间序列或机器学习模型(代理)是根据在聚类级别执行的初始趋势分析来选择的。然后通过强化学习机制对单个组件(智能体)的预测进行权衡,以获得最终预测结果。假设 时间序列模型 , 用于预测,其中 -th 时间步长表示 组件 的预测精度,即组件 给出正确预测的测试周期内的时间分数。
The "correctness" or "accuracy" of a component j's prediction at a time could be a [0,1] variable, equal to 1 if 's prediction is correct (within a pre-defined acceptable error range) for time t; 0 otherwise. It could also depend on the relative accuracy of the prediction, as implemented in the model presented here:
分量 j 的预测在一次 的“正确性”或“准确性”可以是一个 [0,1] 变量,如果 时间 t 的预测是正确的(在预定义的可接受误差范围内),则等于 1;否则为 0。它还可能取决于预测的相对准确性,如此处介绍的模型中所实现的那样:
Recognizing that each predictor is likely to be accurate with some probability, the "consensus" prediction will be a weighted sum of the values , i.e.:
认识到每个预测变量在一定概率下可能是准确的,“共识”预测将是值 的加权总和,即:
where is the weight associated with predictor for the -th interval. The weights account for the respective prediction accuracy fractions and the importance of performance in previous time steps. The weights are normalized and sum to in case ties are allowed, i.e., two components could have the same accuracy at the same time.
其中 ,是与 第 -th 个区间的预测变量 关联的权重。权重考虑了各自的预测准确性分数和先前时间步中性能的重要性。权重被归一化,并在允许平局的情况下求和, 即两个分量可以同时具有相同的精度。
The main feature of RL is the manner in which the weights are adjusted from one time step to next time step based on the latest information derived from actual observed values. The basic idea is that will be set for the next time step based on the outcome of the current time step . One might also prefer positive deviations (Forecast - Actual ) to negative deviations (Forecast Actual ). Note that it is usually better to over-predict than under-predict, as it is possible to redeploy or give discounts to customers in the former case, while customers and goodwill are lost in the latter case. The preference for over-prediction is reflected by including a penalty coefficient in the weights applied to a forecast as follows:
RL的主要特点是根据从实际观测值得出的最新信息,从一个时间步长调整权重到下一个时间步长。基本思想是, 将根据当前时间步 的结果为下一个时间步 设置。人们可能更喜欢正偏差(预测 - 实际 )而不是负偏差(预测实际 )。请注意,高估通常比低估要好,因为在前一种情况下可以重新部署或向客户提供折扣,而在后一种情况下会失去客户和商誉。通过在应用于预测的权重中包含惩罚系数 来反映对过度预测的偏好,如下所示:
where 哪里
: Deviation penalty for component model
:组件模型 的偏差惩罚
: Forecast for period from model
:模型 的周期 预测
Observation at period
经期 观察
: Weight given for forecast from model at period
:根据模型 在期间 进行预测的权重
: Weight given to past data
: Number of fitting (component) models used
: 使用的管接头(元件)型号数量

3.3. Forecasting over different time scales
3.3. 不同时间尺度的预测

As previously mentioned the proposed forecasting scheme is used to support operational planning over the short term and the longer term. The operations are planned on a weekly time scale therefore both short-term and long-term forecasts represent the number of moves per week. Accordingly, the short-term model is designed to predict the number of moves for one week and two weeks in advance whereas the long-term model forecasts the number of moves in each week for the weeks in one month and two

months in advance. 提前几个月。
For the short-term forecast, a suitable number of time series models are chosen for each cluster based on the trend characteristics found during the pre-processing stage. However, time series models have low accuracy when it comes to forecasting over long horizons (Nguyen and Chan, 2004), so an alternative to long-term weekly forecasts is devised. Consequently, instead of directly forecasting weeks far in advance, we forecast the number of moves on a monthly basis, i.e., the number of moves that will occur in the next two months separately. Afterwards, the number of moves in each week is calculated as a weighted average of: (a) No Information/ Equal Allocation (NIA) Model, and (b) Monthly to Weekly Mapping (MWM) Model with the weights ( ) update using an RL mechanism similar to the one previously described. The final long-term weekly forecast is calculated as follows:
对于短期预测,根据预处理阶段发现的趋势特征,为每个聚类选择适当数量的时间序列模型。然而,时间序列模型在长期预测方面的准确性较低(Nguyen和Chan,2004),因此设计了一种长期每周预测的替代方案。因此,我们不是提前几周直接预测,而是按月预测移动次数,即未来两个月将分别发生的移动次数。之后,每周的移动次数计算为以下加权平均值:(a) 无信息/均等分配 (NIA) 模型,以及 (b) 月度到每周映射 (MWM) 模型,权重 ( ) 使用类似于前面描述的 RL 机制进行更新。最终的长期每周预测计算如下:
where 哪里
: Final long-term weekly forecast for week - regardless of the month it falls in
:每周 的最终长期每周预测 - 无论它属于哪个月份
Resultant forecast for week from the "No Information Model"
“无信息模型”对本周 的预测结果
: Resultant forecast for week from the "Monthly to Weekly Mapping model"
:根据“月度到周度映射模型”对周 的预测结果
The RL weight for model and for week
模型 和周 的 RL 权重
In the NIA model, the monthly forecasts are equally allocated over the days of the month. Then the weekly patterns are obtained by adding up the days of the weeks that belong to that month. On the other hand, the MWM model requires inputting a matrix of week-month weights defined as the proportion of days in week that fall in month , the monthly forecasts and the long-term weekly forecasts using time series models . Although time series models are not optimal for long-term forecasting, they are incorporated in this overall model to take advantage of their ability to pick up trends and patterns in the data albeit not with the best accuracy. Afterwards, the monthly forecasts are mapped into the weeks - in each month - using weights as follows:
在 NIA 模型中,月度预测 在当月的几天内平均分配。然后,通过将属于该月的周数相加来获得每周模式。另一方面,MWM 模型需要输入一个周-月权重矩阵, 该矩阵定义为使用时间序列模型 的月 度预测 和长期每周预测的周数百分比。尽管时间序列模型对于长期预测来说不是最佳选择,但它们被合并到这个整体模型中,以利用它们获取数据中趋势和模式的能力,尽管不是最佳精度。之后,使用如下权重 将月度预测映射到每个月的周数中:
where 哪里
: The fraction of the number of moves in month that fall in week
:以周 为单位的月份 移动次数的百分比
: The proportion of days in week that fall in month
:一周 中以月 为单位的天数比例
: The weekly forecast for from the component models
:组件模型的 每周预测
: The forecasted number of moves for month
:当月的 预测移动次数
: The adjusted forecast for week based on the monthly forecasts and the weekly pattern
:根据月度预测和每周模式调整后的周 预测
By definition, is the proportion of days in week that fall in month "i"; for example, the week from December 30,2019 to January 6, 2020 has 2 days out of 7 in December. Therefore, if week does not belong to month , then . It is an important property that ensures , in other words, the number of moves during one month is not over/underestimated. This, in turn, ensures that the sum of the weekly forecasts matches the sum of the monthly forecasts in one year as shown below:
根据定义, 是一周 中落在月份“i”的天数的比例;例如,从 2019 年 12 月 30 日到 2020 年 1 月 6 日的一周,12 月的 7 天中有 2 天。因此,如果 week 不属于 month ,则 。这是一个重要的属性,可以确保 换句话说,一个月内的移动次数不会被高估/低估。这反过来又确保了每周预测的总和与一年内的月度预测总和相匹配,如下所示:
where 哪里

3.4. Accounting for calendar effects
3.4. 考虑日历效应

Holiday shopping season is critical for suppliers and retailers who rely on supply chain companies to deliver their products to the right place at the right time. Therefore, it is necessary to account for holidays as covariates in the time series models whenever possible. Nevertheless, this introduces additional challenges known as the calendar effect.
Holidays such as Easter and Thanksgiving occur each year but their exact timing (such as date for Thanksgiving, day of the week for Christmas) shifts and their effect on the time series may affect more than one period (one week or one month). Therefore, the observations in the time series (in this case the number of container moves) depend on the absolute length of the period between moving holidays. It is important because the standard time series models, such as ARIMA do not capture these lags and effects. These models do capture seasonality, but by definition, the seasonal component occurs at the same time each year, and since these periods and holidays are shifting by date or day of the week, their effect is not picked up by the models.
复活节和感恩节等假期每年都会发生,但它们的确切时间(例如感恩节的日期、圣诞节的星期几)会发生变化,并且它们对时间序列的影响可能会影响多个时期(一周或一个月)。因此,时间序列中的观测值(在本例中为集装箱移动次数)取决于移动假期之间时间段的绝对长度。这很重要,因为标准时间序列模型(如 ARIMA)无法捕获这些滞后和影响。这些模型确实捕获了季节性,但根据定义,季节性成分发生在每年的同一时间,并且由于这些时期和节假日按日期或星期几变化,因此模型不会接收到它们的影响。
Cleveland and Grupe (1981) developed an approach to correct for the calendar effects. The approach used in their paper was developed for monthly forecasts however in this framework forecasts are generated on a weekly basis, so the approach is modified accordingly. Then the calendar effects of different holidays are added as external covariates (regression variables) to the models. The regression variables (effects) of the calendar events are defined as:
where 哪里
: The number of days before the holiday that are affected by it
: The current day
: Number of days in week that belongs to the month of day and are affected by the holiday
:一周 中属于一天 中受假日影响的月份的天数
: Total number of days in week
:一周 的总天数
The holidays included in this study are New Year's, Easter, Memorial Day, Fourth of July, Labor Day, Thanksgiving and Christmas. Their assumed values are listed as follows:
本研究中包括的假期是新年、复活节、阵亡将士纪念日、七月四日、劳动节、感恩节和圣诞节。其假定 值如下所示:
  • New Year's - from Dec. 25th until Jan 2nd
    新年 - 从 12 月 25 日到 1 月 2 日
  • Easter - 10 days
    复活节 - 10 天
  • Memorial Day -7 days
    阵亡将士纪念日 -7 天
  • 4th of July - 7 days
    7 月 4 日 - 7 天
  • Labor Day - 7 days
    劳动节 - 7 天
  • Thanks giving - 10 days
    感恩 - 10 天
  • Christmas is variable, and it is taken as the duration starting from the day after Thanksgiving up to and including Dec 24th.
    圣诞节 是可变的,它被视为从感恩节后的第二天开始到 12 月 24 日(包括 12 月 24 日)的持续时间。

3.5. Evaluation metrics 3.5. 评估指标

Several measures are used for assessing the prediction accuracy of the proposed forecasting method. Both statistical measures such as Mean Absolute Percentage Error (MAPE) and Mean Absolute Deviation (MAD) are utilized. Additionally, the results are interpreted graphically using a graph of error frequency distribution as well as individual weekly and cumulative forecasts. The statistical measures are presented below:
本文采用多种措施评估所提预测方法的预测精度。同时使用平均绝对百分比误差 (MAPE) 和平均绝对偏差 (MAD) 等统计度量。此外,使用误差频率分布图以及单个每周和累积预测以图形方式解释结果。统计指标如下:
where 哪里
actual number of moves during period
期间 的实际移动次数
forecasted number of moves for period
期间 的预测移动次数
While MAPE measures the extent of the error in percentage units, the MAD measures the error in units of the object forecasted.
MAPE 以百分比单位测量误差程度,而 MAD 以预测对象的单位测量误差。
Error frequency distribution graphs allow examination of the performance of the model. Ultimately, a bell-shaped distribution is preferred such that it is centered on 0% MAPE with most of the errors falling in between . Moreover, the individual weekly
误差频率分布图允许检查模型的性能。最终,钟形分布是首选,因此它以 0% MAPE 为中心,大多数误差介于两者之间 。此外,个人每周
Fig. 3. Clustering market at the lane level.
图 3.在车道层面聚集市场
graph helps examine how closely the forecasts follow the actual observation patterns as well as their proximities to the real values. As for the cumulative graphs, these are used for the long-term weekly pattern model to ensure that the forecasts are catching up with the actual number of moves and therefore to check and verify the soundness of the proposed approach.

4. Case study 4. 案例研究

The proposed approach is tested on a dataset provided by an intermodal company operating in the USA. The dataset includes information for each move by origin and destination. Moreover, for each movement, the dataset has information regarding the commodity type and order placement, pickup and delivery times. The data used in the model is the weekly number of moves from 2013 to 2017. The information presented below is for a market referred to here as X (for data confidentiality purposes).
所提出的方法在一家在美国运营的多式联运公司提供的数据集上进行了测试。该数据集包括按始发地和目的地划分的每次移动的信息。此外,对于每个移动,数据集都包含有关商品类型和订单下达、取货和交货时间的信息。模型中使用的数据是 2013 年至 2017 年的每周移动次数。下面提供的信息适用于此处称为 X 的市场(出于数据保密目的)。
After pre-processing the dataset, the moves originating from market are clustered by lane then each lane group is further clustered by commodity type. The clustering at the lane level results in three groups with the members of each cluster- destinations presented in green, blue and black respectively and the centroids are chosen by the k-means algorithm in red, shown in Fig. 3. Figs. 3 and 4 only show the first 52 weeks in the training set - for clear presentation. However, clustering was done using the whole training set. The plot shows that the grouping is mainly driven by the frequency of moves between market and the different destinations. Accordingly, three clusters are referred to as the "More Frequent" in green, "Frequent" in blue, and "Less Frequent" in black with cluster size of one, four and six lanes respectively. Through Fig. 3, it is clear that the "More Frequent" group is distinct whereas the other two groups have some overlapping elements.
在对数据集进行预处理后,源自市场的 移动按车道进行聚类,然后按商品类型进一步对每个巷组进行聚类。在车道层面进行聚类会产生三组,每个聚类的目标的成员分别以绿色、蓝色和黑色表示,质心由红色的 k 均值算法选择,如图 3 所示。图 3 和图 4 仅显示训练集中的前 52 周 - 以便清晰显示。但是,聚类是使用整个训练集完成的。该图显示,分组主要由市场 与不同目的地之间的移动频率驱动。因此,三个集群分别以绿色表示“较频繁”,以蓝色表示“频繁”,以黑色表示“不太频繁”,簇大小分别为一、四和六车道。通过图3,可以清楚地看出,“更频繁”的组是不同的,而其他两个组有一些重叠的元素。
Fig. 4. Clustering market at the commodity level.
图 4.在大宗商品层面聚集市场

Lane Group Clustering 车道组聚类

Fig. 5. Weekly moves by cluster.
图 5.每周按集群移动。
Commodities, categorized based on a two-digit code, are clustered based on the number of weekly orders, as shown in Fig. 4. Starting with the original dataset, the number of weekly moves for 2-digit commodity ID is clustered. This results in two distinct groups presented in blue and green with cluster size two and 23 commodities respectively. Fig. 4 shows that there is no overlap between two clusters and that the primary distinction is again the number of weekly moves.
商品,根据两位数代码分类,根据每周订单数量进行聚类,如图 4 所示。从原始数据集开始,对 2 位商品 ID 的每周移动次数进行聚类。这导致两个不同的组以蓝色和绿色显示,集群大小分别为 2 和 23 种商品。图 4 显示两个集群之间没有重叠,主要区别再次是每周移动的次数。
Fig. 5 demonstrates the two main steps in the clustering process. The blue numbers on the arrows indicate the average number of weekly moves by cluster.
图 5 演示了聚类过程中的两个主要步骤。箭头上的蓝色数字表示按集群划分的每周平均移动次数。
As a result, the model is customized for each lane-commodity cluster, in other words, each lane cluster is divided into two commodity sub-sets, and the model is designed based on the specifications of each. The number of moves from January 2013 to June 2017, is extracted from the dataset and then aggregated to weekly moves for the short-term model and monthly moves for the longterm model. Afterwards, the number of container moves is divided into two sets: the "Training Dataset" is used for estimation from January 2013 to June 2015, and the "Testing Dataset" is predicted and introduced in the model gradually using the rolling horizon approach.
因此,针对每个车道商品集群定制模型,即将每个车道集群划分为两个商品子集,并根据每个子集的规格设计模型。从数据集中提取 2013 年 1 月至 2017 年 6 月的移动次数,然后聚合为短期模型的每周移动数和长期模型的每月移动数。然后,将集装箱移动次数分为两组:使用“训练数据集”进行2013年1月至2015年6月的估计,并使用滚动水平方法逐步将“测试数据集”引入模型中。

4.1. Short-term model 4.1. 短期模型

This model is used for forecasting one week and two weeks in advance. A "hybrid" model is used for forecasting such that the is forecasted by combining data from lane clusters two and three into one cluster and forecasting for it. The "hybrid" approach was compared to forecasting the separate clusters and the combination, in this case, resulted in a higher prediction accuracy. Therefore, we decided to combine the two clusters for this case only. On the other hand, forecasts are generated using three lane clusters previously discussed. The hybrid approach was introduced after the non-hybrid approach failed to produce acceptable results.
该模型用于提前一周 和两周 进行预测。“混合”模型用于预测, 通过将来自车道集群 2 和 3 的数据合并到一个集群中并对其进行预测来预测。将“混合”方法与预测单独的聚类进行了比较,在这种情况下,这种组合导致了更高的预测准确性。因此,我们决定仅针对这种情况合并两个集群。另一方面, 预测是使用前面讨论的三个车道集群生成的。混合方法是在非混合方法未能产生可接受的结果之后引入的。
The time series components used in forecasting are: autoregressive integrated moving average with explanatory variables (ARIMAX) with the holiday covariates, Seasonal and Trend decomposition using Loess (STL), and Simple Moving Average (SMA). The components used in forecasting are ARIMAX, TBATs, and SMA. In the ARIMAX models, the dependent variable is regressed on its own lagged values and lagged forecasting errors as well as covariates. The model may also apply an initial step of differencing to prevent non-stationarity. STL identifies the seasonal, trend and irregular components using locally weighted smoothing (local regression). SMA forecasts are calculated as the average of some previous observations (Hyndman and Athanasopoulos, 2014). Moreover, TBATs is a generalization of exponential smoothing. It allows for automatic Box-Cox transformation and errors are modeled as an ARMA process (De Livera et al., 2011).
预测中使用的时间序列分量 包括:具有解释变量的自回归综合移动平均线 (ARIMAX) 和假日协变量、使用黄土的季节性和趋势分解 (STL) 以及简单移动平均线 (SMA)。预测中使用 的组件是 ARIMAX、TBAT 和 SMA。在 ARIMAX 模型中,因变量根据其自身的滞后值和滞后预测误差以及协变量进行回归。该模型还可以应用微分的初始步骤来防止非平稳性。STL 使用局部加权平滑(局部回归)识别季节性、趋势和不规则分量。SMA预测被计算为一些先前观测值的平均值(Hyndman和Athanasopoulos,2014)。此外,TBAT是指数平滑的推广。它允许自动Box-Cox转换,并将错误建模为ARMA过程(De Livera等人,2011)。
The training period is taken as 2.5 years and is kept so during the entire process of adding and dropping weeks. The initial forecasts are generated using a three-year period; however using 2.5 years improves the accuracy of results, which may be related to Erev and Roth's (1998) concept of recency discussed in Section 2. Finally, the forecasts are generated at the lane-commodity cluster
培训期为 2.5 年,并在增加和减少周数的整个过程中保持如此。初步预测是以三年为周期生成的;然而,使用 2.5 年可以提高结果的准确性,这可能与第 2 节中讨论的 Erev 和 Roth (1998) 的新近概念有关。最后,在车道-商品集群上生成预测
Fig. 6. Aggregate Short Term Forecasts for Market X - Forecasts starting 2017.
图 6.市场 X 的汇总短期预测 - 从 2017 年开始的预测。
Fig. 7. MAD in forecasting for and for market .
图 7.MAD在市场预测 市场 预测中。
level with positive deviation penalty equal to the negative deviation penalty, shown in Fig. 6 . The evaluation metrics, MAD and MAPE, in Figs. 7 and 8 are presented by one-week ahead forecasts and two-weeks ahead forecasts indicated on the axis. The figures also present the metrics as an average of all testing time intervals (All) and testing time intervals in 2017 only (2017). The approach proposed result in forecasts with a margin of error of 20 weekly moves on average (Fig. 7), and of for the overall forecasting period and for 2017 forecasts only (Fig. 8). The errors for ( 2 weeks ahead) forecasts increase to 50 weekly moves on average and around 12-16% MAPE (Figs. 7 and 8). The MAD for one and two weeks ahead are reasonable given that the demand to the market fluctuates around 450 moves per week, Fig. 6.
正偏差惩罚等于负偏差惩罚的水平,如图 6 所示。图 7 和图 8 中的评估指标 MAD 和 MAPE 由 轴上 指示的提前一周预测 和提前两周预测表示。这些数字还以2017年(2017年)所有测试时间间隔(全部)和测试时间间隔的平均值表示指标。该方法提出的 预测结果平均每周波动幅度为20次(图7),并且仅 针对整个预测期和 2017年的预测(图8)。 (提前 2 周)预测的误差平均增加到 50 周,MAPE 约为 12-16%(图 7 和 8)。鉴于市场需求每周波动约 450 次,未来 1 周和 2 周的 MAD 是合理的,图 6。
Analyzing the error distributions for forecasts (Fig. 9) shows that the relative errors follow a favorable normal distribution centered on zero. errors (Fig. 9) are more spread out, but the majority of the forecasting errors are concentrated around zero.
分析 预测的误差分布(图9)表明,相对误差遵循以零为中心的有利正态分布。 误差(图9)更加分散,但大多数预测误差集中在零附近。
Figs. 10 and 11 show the cluster level forecasts for one week ahead. The forecasted values closely follow the actual observations and trends in demand. It is further verified by the low errors presented in Figs. 12 and 13. On the other hand, forecasts for two weeks ahead catch up with the fluctuations at a slower pace (Figs. 14 and 15). This is reflected in the errors in Figs. 16 and 17.
图10和图11显示了未来一周的集群水平预测。预测值与实际观察结果和需求趋势密切相关。图 12 和图 13 中的低误差进一步验证了这一点。另一方面,对未来两周的预测以较慢的速度赶上波动(图14和图15)。这反映在图 16 和 17 中的误差中。
Two-week ahead forecasts for the remaining clusters are still performed separately as discussed before. Below, their forecasts are summed and presented as forecasts for "Lane Cluster 2" to maintain a consistent format with forecasts.
如前所述,对其余集群的两周前期预报仍单独进行。下面,他们的预测被汇总并呈现为“Lane Cluster 2”的预测,以保持与 预测一致的格式。

4.2. Long-term model 4.2. 长期模型

As previously mentioned, the long-term model combines the forecasts of the No Information model and the Monthly to Weekly Mapping model.
The monthly model forecasts the total number of moves that will occur in the month or where is the current month. The forecasts are generated using the hybrid approach in which lane clusters 1 and 2 are combined. For that cluster, STL and structural time series (StructTS) are used with rolling history of length 2.5 years. Forecasts are generated at the lane commodity level and penalizing the negative deviations. As for the third lane cluster, all the specifications apply except for the components, which consist of SMA, STL, complex exponential smoothing (CES), and StructTS. StructTS is a linear-state space model that decomposes the time series into components (trend, seasonality, etc.) and fits the model by maximum likelihood (Petris and Petrone, 2011). CES is a nonlinear method that introduces complex variables to simple exponential smoothing method (Svetunkov et al., 2016).
月度模型预测当月 将发生的移动总数或 当月的位置 预测是使用混合方法生成的,其中车道集群 1 和 2 被组合在一起。对于该聚类,使用 STL 和结构时间序列 (StructTS),滚动历史长度为 2.5 年。预测是在车道商品层面生成的,并对负偏差进行惩罚。对于第三通道集群,除组件外,所有规格均适用,这些组件由 SMA、STL、复指数平滑 (CES) 和 StructTS 组成。StructTS 是一种线性状态空间模型,它将时间序列分解为组件(趋势、季节性等),并以最大似然拟合模型(Petris 和 Petrone,2011 年)。CES是一种非线性方法,它将复杂变量引入简单的指数平滑方法(Svetunkov等人,2016)。
Separate forecasts are generated for each lane cluster. The components for lane cluster 1 include TBATs and CES. Lane cluster 2 forecasts are generated using STL and CES whereas lane cluster 3 forecasts are generated using SMA, STL, CES, and StructTS. The previous specifications still apply.
为每个车道集群生成单独的 预测。泳道集群 1 的组件包括 TBAT 和 CES。车道集群 2 预测使用 STL 和 CES 生成,而车道集群 3 预测使用 SMA、STL、CES 和 StructTS 生成。以前的规范仍然适用。
Fig. 8. MAPE in forecasting for and for market .
图 8.MAPE在市场预测 市场 预测中。

Fig. 9. Error Distribution in short term forecasts in Red, in Green) for aggregate market . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
图 9.短期预测 中的误差分布(红色, 绿色)为总市场 。(为了解释此图例中对颜色的引用,读者可以参考本文的网络版本。
Fig. 10. Forecasts for market - Cluster 1 .
图 10. 市场 预测 - 集群 1 .
Fig. 11. Forecasts market - hybrid Cluster 2 .
图 11. 预测市场 - 混合集群 2 .
Fig. 12. MAD in forecasting for for Lane Cluster 1 and 2- market .
图 12.MAD 对 Lane Cluster 1 和 2- 市场的 预测
Fig. 13. MAPE in forecasting for for Lane Cluster 1 and 2- market .
图 13.MAPE 对 Lane Cluster 1 和 2 市场的 预测
Fig. 14. Forecasts market - Lane Cluster 1 .
图 14. 预测市场 - Lane Cluster 1 .
Based on the results presented above (Figs. 18-21), it is evident that the monthly forecasts are accurate enough to be incorporated into the long-term weekly model. The calculations presented next are still performed within a rolling horizon framework. First, the weekly patterns are obtained for the "No Information / Equal Allocation" component of the model. It is done by an equal allocation of the monthly forecasts over the days of the month. Then, for each week, the number of daily moves that belong to that week and month are added.
Afterwards, the "Monthly to Weekly Mapping" component is calculated as a function of a week-month weights matrix , forecasts , and the long-term weekly forecasts using time series models . The component models used for the long-term
之后,“月度到周度映射”组件被计算为周月权重矩阵 、预测 和使用时间序列模型 的长期每周预测的函数。用于长期的组件模型
Fig. 15. Forecasts for market - hybrid Lane Cluster 2 .
图 15. 市场 预测 - 混合车道集群 2 .
Fig. 16. MAD in forecasting for