Analysis of the Mathorcup C Problem in Mathematical Modeling 2024 – Mathematical Modeling AI

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Mathorcup C Problem

📖 Problem Background：

• The e-commerce logistics network consists of multiple links in order fulfillment. The sorting center, as an intermediate link in the network, needs to sort packages according to different flows and send them to the next location, ultimately delivering the packages to consumers. Improving the management efficiency of the sorting center plays a crucial role in the overall network’s fulfillment efficiency and operational costs. The volume forecasting of the sorting center is an important research issue in the e-commerce logistics network, and accurate forecasting of the sorting center’s volume is the basis for subsequent management and decision-making. The volume forecasting objectives generally have two aspects: first, to predict the daily volume of each sorting center based on historical volume, logistics network configuration, and other information; second, to predict the hourly volume of each sorting center based on historical hourly data. The volume forecasting of the sorting center is related to the transportation routes of the network. When the route relationships are adjusted, the adjustment information can be referenced to obtain a more accurate forecast of the volume at each sorting center. Personnel scheduling based on the volume forecast of the sorting center is also an important issue. The personnel at the sorting center include both regular and temporary workers, with regular workers being more efficient and temporary workers having relatively lower efficiency and higher hiring costs. This logistics network includes 57 sorting centers, and the daily volume of each sorting center over the past four months is shown in Attachment 1, while the hourly volume over the past 30 days is shown in Attachment 2.

📖 Extracted Questions：

• 📋 Question 1
• Establish a volume forecasting model to predict the daily and hourly volume of the 57 sorting centers for the next 30 days, and write the forecast results into Result Tables 1 and 2.
• 📋 Question 2
• The average volume of each transportation route between sorting centers over the past 90 days is shown in Attachment 3. If the transportation routes between sorting centers change in the next 30 days, as specified in Attachment 4, please predict the daily and hourly volume of the 57 sorting centers for the next 30 days based on Attachments 1-4, and write the forecast results into Result Tables 3 and 4.
• 📋 Question 3
• Assuming each sorting center has 60 regular workers, priority will be given to using regular workers in personnel arrangements. If additional personnel are needed, temporary workers will be used. Based on the forecast results of Question 2, establish a model to provide the attendance numbers for each shift at each sorting center for the next 30 days, and write the results into Result Table 5. The total number of person-days arranged (for example, if 200 employees attend each day for 30 days, the total person-days would be 6000) should be minimized while ensuring that the actual hourly efficiency is as balanced as possible.
• 📋 Question 4
• Study the scheduling problem of a specific sorting center, taking SC60 as an example. Assuming that sorting center SC60 currently has 200 regular workers, based on the forecast results of Question 2, establish a model to determine the attendance plan for each regular and temporary worker for the next 30 days, specifying which shifts each regular worker will attend and how many temporary workers need to be hired for each shift, and write the results into Result Table 6. The attendance rate of each regular worker (the number of days attended divided by the total number of days, 30) cannot exceed 85%, and the number of consecutive working days cannot exceed 7 days. The total number of person-days arranged should be minimized while ensuring that the actual hourly efficiency is as balanced as possible, and the attendance rate of regular workers should also be as balanced as possible.

Problem Analysis

• The four questions form a closely linked progression chain, presenting a logical sequence of “basic forecasting → network adjustment → sorting center-level scheduling → employee-level scheduling.” The output of the previous question serves as the input for the next question, and the constraints become progressively tighter as the questions advance. Specifically, Question 1 establishes a baseline forecasting model, providing raw volume data support for all subsequent questions; Question 2 introduces network dynamics based on Question 1, considering the volume redistribution caused by route adjustments; Question 3 conducts overall manpower planning for the sorting center based on the adjusted volume from Question 2, addressing the conflict between resource efficiency and cost; Question 4 refines the scheduling down to individual employees within the framework of Question 3, adding constraints for personnel management. Each question is tightly coupled through data flow and constraint transmission, where forecasting errors propagate through the levels affecting scheduling quality, while changes in network structure simultaneously impact volume forecasting and personnel scheduling strategies.

🤯 Question 1：

• Background and Internal Connections with Other Questions：In the e-commerce logistics network, volume forecasting at sorting centers is the foundation for subsequent management and decision-making. Accurately predicting the daily and hourly volume of each sorting center helps in resource allocation, improving operational efficiency, and reducing costs. Question 1 is the basis of the entire problem chain, and its forecast results provide data support for Questions 2, 3, and 4. Question 2 builds on Question 1 by considering changes in transportation routes, while Questions 3 and 4 optimize personnel scheduling based on the forecast results of Question 2.

Detailed Explanation of the Answering Approach from Multiple Perspectives:

• Influencing Factors：
• Time Dimension：There are daily cycles (e.g., weekend effects, where weekend volumes may differ from weekdays), weekly cycles (e.g., promotional cycles, where volumes in specific weeks may increase due to promotional activities), and holiday effects (volumes typically fluctuate significantly during holidays).
• Spatial Dimension：Different sorting centers have different locational characteristics; for example, some sorting centers may serve as regional hubs, resulting in relatively larger volumes and different patterns of change.
• Data Characteristics：Four months of daily data can provide medium- to long-term trends, while 30 days of hourly data can capture daily fluctuations in volume.
• Theoretical Basis：Involves time series analysis theories, such as the ARIMA model, which captures linear characteristics of time series based on autoregression, differencing, and moving averages; the LSTM model, based on deep learning, is suitable for handling time series data with long-term dependencies. Additionally, relevant knowledge from statistics and machine learning is applied.
• Model Construction：Adopt a hierarchical forecasting architecture.
• Daily Forecasting：Choose a model combining Prophet (which can handle multiple seasonalities) and LightGBM (which can perform feature engineering). Feature design includes temporal features (e.g., lag variables, moving window statistics), external features (holiday markers, weekday factors), and spatial features (sorting center category encoding, such as grading based on historical volume). The training strategy is to model independently by sorting center to avoid interference from spatial heterogeneity.
• Hourly Forecasting：Use the STL decomposition strategy, taking the daily forecast results as the trend component, extracting daily cycle patterns from historical hourly data, and adding residual components to capture special fluctuations. Introduce an Attention-LSTM network to correct distributions during special periods such as holidays.
• Algorithm Selection：The example code uses Prophet for daily forecasting, preprocessing the data, constructing the model, training the model, and generating forecast results for the next 30 days.

Considerations During the Answering Process:

• When preprocessing data, pay attention to handling data sparsity and outliers. Data sparsity can be alleviated through data augmentation, feature engineering, etc., such as using interpolation to fill in missing data and extracting more features to enrich model input; outlier handling can use statistical methods based on standard deviation to identify and remove them, or use robust models to reduce their impact.
• For parallel forecasting across multiple sorting centers, the choice between establishing independent models or a unified model should be weighed based on actual conditions. Independent models can better adapt to the unique patterns of each sorting center but require more computation; unified models are more efficient but may not fully capture the differences between sorting centers.
• In the collaboration between daily and hourly forecasting, ensure the accuracy of the historical hourly volume ratio to reasonably decompose the daily total into hourly volumes.

🤯 Question 2：

• Background and Internal Connections with Other Questions：In actual e-commerce logistics networks, transportation routes may change, which can affect the volume at sorting centers. Therefore, it is necessary to make more accurate predictions of the volume at each sorting center while considering route changes. Question 2 expands on Question 1 by considering the factor of transportation route changes. Its forecast results also provide data support for Questions 3 and 4, influencing subsequent personnel scheduling optimization.

Detailed Explanation of the Answering Approach from Multiple Perspectives:

• Influencing Factors：Mainly the changes in transportation routes, including the addition and removal of routes, which will lead to a redistribution of volume among different sorting centers.
• Theoretical Basis：Based on graph theory and network analysis theory, treating sorting centers as nodes and transportation routes as edges, quantifying the impact of route changes on volume by analyzing the network structure and flow relationships between nodes.
• Model Construction：
• Construct a volume transfer matrix, defined as the average daily volume from SC_i to SC_j (from Attachment 3). When routes change, calculate the inflow/outflow changes for each node, such as,.
• Use a forecasting correction model, whereis the network sensitivity coefficient, determined by fitting historical data.
• Algorithm Selection：When quantifying the impact of route changes on volume, it is necessary to calculate the average volume for each route based on historical data and adjust the corresponding sorting center’s volume according to the addition or removal of routes. For volume allocation of added/removed routes, reference historical averages or similar routes, such as using cosine similarity to find alternative paths.

Considerations During the Answering Process:

• When quantifying the impact of route changes on volume, ensure the accuracy and representativeness of historical data to accurately calculate the adjustments in inflow/outflow.
• When determining the network sensitivity coefficient, fit historical data using reasonable methods to avoid inaccuracies in the coefficient leading to deviations in forecast results.
• When handling volume allocation for added/removed routes, ensure that the selection of similar routes is reasonable and that their volume distribution has reference value.

🤯 Question 3：

• Background and Internal Connections with Other Questions：Given the known daily and hourly volume for each sorting center over the next 30 days, it is necessary to arrange personnel attendance reasonably to meet volume processing needs while reducing costs and improving efficiency. Question 3 optimizes scheduling at the sorting center level based on the forecast results of Question 2. Its results provide a foundational framework for Question 4, which further refines scheduling at the employee level.

Detailed Explanation of the Answering Approach from Multiple Perspectives:

• Influencing Factors：Mainly the volume demand of each sorting center and the efficiency and cost differences between regular and temporary workers.
• Theoretical Basis：Based on integer programming theory, establishing a mathematical model to describe the objectives and constraints of the problem, seeking the optimal manpower allocation scheme.
• Model Construction：
• Decision Variables：Letbe the number of regular workers for shift s on day d, andbe the number of temporary workers for shift s on day d.
• Objective Function：whereis the hourly efficiency (volume/(number of people × efficiency)),is the balance weight.
• Constraints：Include volume coverage, upper limit for regular workers, and non-negative integer constraints.
• Algorithm Selection：Use the Gurobi solver to handle mixed-integer quadratic programming problems. To speed up the solution, relax the temporary worker variables to continuous variables.

Considerations During the Answering Process:

• When converting volume into manpower demand, accurately set the efficiency coefficients for regular and temporary workers to ensure that the calculated manpower numbers are reasonable.
• When determining the balance weight, weigh the importance of total person-days and efficiency balance based on actual conditions to avoid inappropriate weight settings leading to results deviating from expectations.
• Integer programming solutions are complex; when using the Gurobi solver, pay attention to parameter settings, and consider the impact of relaxing temporary worker variables to continuous variables on the results.

🤯 Question 4：

• Background and Internal Connections with Other Questions：At a specific sorting center (e.g., SC60), it is necessary to consider individual constraints for employees, such as attendance rates and continuous working limits, to conduct more refined scheduling optimization. Question 4 is an extension of Question 3, adding employee-level constraints based on the results of Question 3. Its results provide the final optimized scheduling plan for personnel at a specific sorting center.

Detailed Explanation of the Answering Approach from Multiple Perspectives:

• Influencing Factors：In addition to volume demand and differences in personnel efficiency and costs, individual constraints for regular workers, such as attendance rates not exceeding 85% and continuous working days not exceeding 7 days, are also considered.
• Theoretical Basis：Based on combinatorial optimization theory and multi-objective optimization theory, searching through large combinatorial spaces to find optimal scheduling solutions that meet multiple objectives and constraints.
• Model Construction：
• Decision Variables：Letindicate whether employee e attends shift s on day d.
• Objective Function：.
• Hard Constraints：Attendance days, continuous work, shift coverage.
• Algorithm Selection：Design an improved NSGA-II algorithm, using a three-dimensional binary matrix (employees × dates × shifts) for encoding, with genetic operations including crossover (block exchange to retain feasible solutions) and mutation (intelligent flipping of the least violating genes), and elite retention (archiving Pareto front solutions).

Considerations During the Answering Process:

• When considering individual constraints, ensure that the constraints in the model accurately reflect the actual situation to avoid conflicts that lead to no solution.
• When handling large combinatorial spaces, heuristic algorithms can quickly search for near-optimal solutions, but attention should be paid to parameter settings and convergence to ensure the quality of results.
• When conflicts in constraints arise, reasonably adjust the priority of constraint conditions or adopt relaxed constraints, such as appropriately relaxing certain constraints during peak periods to meet manpower needs, but be careful not to relax too much, leading to results that do not meet actual requirements.

Abstract

• This article focuses on the study of volume forecasting and personnel scheduling issues at sorting centers in the e-commerce logistics network. By establishing volume forecasting models and personnel scheduling models, and employing corresponding solution methods, accurate forecasting of future volumes for 57 sorting centers and reasonable personnel scheduling arrangements are achieved.
• For Question 1, the key point is to predict future volumes based on historical volume data. The approach is to comprehensively consider influencing factors such as time dimensions (daily, weekly, holiday cycles), spatial dimensions (locational characteristics of different sorting centers), and data characteristics (medium- to long-term trends and daily fluctuations), assuming that historical data can reflect future patterns and that volume changes at each sorting center are independent. Based on this, the N-HiTS algorithm is selected, which decomposes time series through multi-rate downsampling, independently predicts each component using lightweight neural networks, and then fuses the results through hierarchical interpolation. During solving, data is prepared and preprocessed, model parameters are determined, time series decomposition, component prediction, and result fusion are performed, with mean squared error as the objective function, and the stochastic gradient descent algorithm is used to train the model, ultimately completing volume forecasting and outputting results.
• The key for Question 2 lies in considering the impact of transportation route changes on volume forecasting. The main influencing factors include the addition and removal of transportation routes, historical volume data, and the relationship between route volumes. Under non-negative volume and data consistency constraints, it is assumed that historical trends continue and that route changes are independent. The TIGER algorithm is used to first construct a volume transfer matrix, calculate the volume changes caused by route changes, and, based on the baseline forecast values from Question 1, determine the network sensitivity coefficients and historical volume statistics, establishing a forecasting correction model, and adjusting through a time-aware graph attention layer. The solving steps include data preparation, matrix construction, volume change calculation, parameter determination, model establishment and adjustment, ensuring that the forecasted volume is non-negative before outputting results.
• Question 3 focuses on optimizing scheduling at the sorting center level based on the forecast results of Question 2. Influencing factors include volume demand, personnel efficiency, and cost differences. Constraints include volume coverage, upper limits for regular workers, and non-negative integer constraints, while assuming stable work efficiency and independent shifts. The Benders-ADMM algorithm is employed, with the main problem optimizing total person-days and shift allocation, and the subproblem solving regular worker scheduling in parallel in ADMM form, accelerating convergence of the main problem through dual information. During solving, data is prepared and initialized, then iteratively solving the main and subproblems, generating valid inequalities, and judging convergence before outputting results.
• Question 4 targets a specific sorting center SC60, considering individual constraints for regular workers based on the forecast results of Question 2 to conduct more refined scheduling optimization. Influencing factors, in addition to volume, efficiency, and cost, also include constraints on regular workers’ attendance rates and continuous working days. Constraints include volume coverage, regular worker attendance, and non-negative integer constraints, also assuming stable work efficiency and independent shifts. The CPO-MOEA algorithm is used, setting the attendance rate limit as a hard constraint and the continuous working days limit as a soft constraint with a penalty weight. During solving, data is prepared, the population is initialized, fitness values are calculated, rapid non-dominated sorting and reference point selection are performed, new solutions are generated through crossover and mutation operations, and the population is merged for environmental selection, outputting results after meeting termination conditions.
• In summary, the models established in this article demonstrate effectiveness in volume forecasting and personnel scheduling, but there are limitations such as dependence on parameters and discrepancies between assumptions and reality. Future improvements can be made through parameter optimization, considering dynamic factors, and enhancing algorithm performance. The model has potential application value in the operational management of e-commerce logistics networks, helping to improve logistics efficiency and reduce costs.
• Keywords：Volume forecasting model, personnel scheduling model, N-HiTS algorithm, TIGER algorithm, Benders-ADMM algorithm, CPO-MOEA algorithm