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1 Introduction

1.1 Background

南部地区食品银行（FBST）的移动食品储藏室（MFP）项目，是纽约州六县经济困难群体获取营养食品的重要保障，2019年已实现70个常规站点、722次年度服务的成熟规模。肺炎疫情对全球食品体系的冲击暴露了供应链与服务体系的脆弱性，尽管不同地区抗冲击能力存在差异，但 FBST 的 MFP 项目仍因疫情导致服务范围大幅收缩、服务模式被迫调整。随着疫情防控形势趋稳，FBST计划2021年恢复疫情前服务水平，取消提前登记要求。如何基于2019年历史数据，制定兼顾需求匹配、公平性与实操性的站点访问调度方案，同时适配天气因素影响与志愿者工作优化需求，成为保障食品援助高效落地、切实满足群众需求的关键问题。

1 The Mobile Food Pantry (MFP) program, operated by the Food Bank of the Southern Tier (FBST), serves as a critical lifeline for food-insecure populations across six counties in New York State. By 2019, the program had achieved significant operational maturity, maintaining a network of 70 regular sites and conducting 722 annual distributions. However, the global shock of the COVID-19 pandemic exposed profound vulnerabilities within food supply chains and service delivery frameworks. While regional resilience varied, the pandemic forced the FBST to significantly contract its service coverage and overhaul its operational models.

As the public health situation stabilizes, FBST aims to restore its service capacity to pre-pandemic levels in 2021, including the removal of pre-registration requirements. The central challenge now lies in leveraging 2019 historical data to design a robust site-visit scheduling scheme. This scheme must achieve a delicate balance between demand alignment, social equity, and operational feasibility, while simultaneously accounting for the stochastic nature of weather conditions and the optimization of volunteer resources. Addressing these complexities is essential for ensuring the efficient delivery of food assistance and effectively meeting the heightened needs of the community.

2 The Mobile Food Pantry (MFP) program, operated by the Food Bank of the Southern Tier (FBST), serves as a vital safeguard for food-insecure populations across six counties in New York State . By 2019, the program had achieved a mature operational scale, encompassing 70 regular sites and conducting 722 annual visits. However, the global food system shocks triggered by the COVID-19 pandemic exposed systemic vulnerabilities in supply chains and service delivery frameworks. Despite varying levels of resilience across different regions, the FBST’s MFP program was forced to significantly contract its service scope and undergo mandatory operational adjustments .

As the pandemic stabilizes, FBST intends to restore its pre-pandemic service levels in 2021 and eliminate the requirement for advance registration. Consequently, a critical challenge has emerged: how to leverage 2019 historical data to design a site visitation schedule that harmonizes demand fulfillment, equity, and operational feasibility . This solution must also integrate considerations for weather-related fluctuations and the optimization of volunteer management to ensure that food assistance is delivered efficiently and effectively to meet the genuine needs of the community.

1.2 Restatement of the problem

考虑到问题陈述中明确的背景信息与限制条件，我们需要解决以下问题：

依据70个常规站点周边社区总需求，制定有效且公平的2021年访问时间表，平均服务所有客户并避免服务差异过大；
基于寒冷冬日客户到访量低、天气好转后附近站点需求激增的历史数据，从 “减少服务站点总数并优化位置”“保持站点数量和位置不变调整访问时间” 中择一，修改原有调度方法，并量化性能改进；
针对同一卡车单次行程能访问两个不同站点的新选项，设计算法选择站点、确定日期及首个站点食物分发量，描述分配的有效性和公平性；
撰写1页执行摘要，阐述所提建议的主要优势和潜在缺点。 Based on the context provided and the requirements for the MCM/ICM competition, here is the professional translation of the "Problem Restatement" section:

1.2 Problem Restatement

Given the background information and constraints provided by the Food Bank of the Southern Tier (FBST) , our team is tasked with achieving the following objectives to optimize the 2021 Mobile Food Pantry (MFP) operations:

• Propose an effective and fair visitation schedule for all 70 regular sites in 2021, ensuring that the frequency of visits is informed by the total demand in surrounding communities to serve all clients well on average and minimize service disparities.

• Modify the previous scheduling approach to incorporate one of two strategic options—either reducing the total number of serviced sites with optimized locations or maintaining existing site locations while adjusting visitation timing—to account for historical weather-related demand fluctuations and quantify the resulting performance improvements.

• Develop an algorithm for the multi-site visitation model (one truck serving two different sites per trip) to determine site selection, scheduling dates, and the specific volume of food to dispense at the first site, while characterizing both the effectiveness and fairness of the resulting distribution.

• Compose a one-page executive summary that articulates the primary advantages and potential drawbacks of the proposed recommendations to support the FBST leadership in their decision-making process.

1.2 Problem Restatement Considering the background information and constraints specified in the problem statement, we are tasked with addressing the following four objectives:

Develop an effective and fair 2021 visitation schedule based on the total community demand surrounding the 70 regular sites, ensuring that all clients are served on average and significant service disparities are minimized.

Modify the existing scheduling approach by selecting one of two strategies： 'reducing serviced sites while optimizing locations' or 'maintaining current sites while adjusting visitation timing', based on historical weather-driven demand fluctuations

Design an algorithm for the one-truck-two-sites operational model, determining site pairings, scheduling dates, and initial site food distribution volume, and evaluate the effectiveness and equity of the distribution

Compose a one-page executive summary that articulates the primary advantages and potential limitations of the proposed recommendations for the Food Bank's leadership.

1.3 Literature review

1.4 Our work

4. Model I: Bi-Objective Site Visit Frequency Allocation and CP-SAT Periodic Scheduling Model双目标站点访问频次分配与 CP-SAT 周期排班模型

4.1 Model Overview

针对 FBST 公益救济项目站点访问规划需求，本模型以 “公平优先、兼顾效率” 为核心，构建 “频次分配 - 效能校验 - 周期排程” 三阶段递进方案，形成从理论到实操的完整闭环，具体思路如下：首先，构建双重公平导向频次分配机制，优先以 kbase 保障所有站点底线服务（覆盖公平），再按站点历史需求占比分配剩余运力（需求适配公平），生成初始频次 kie。其次，通过 “有效性评分 + 基尼系数” 双维度校验筛选最优 kbase：以供需匹配度与失衡惩罚构建有效性模型，分析 kbase 对效能的影响；引入基尼系数（G<0.2 为阈值）衡量站点间效能均衡性，确定最终年度频次。最后，构建 CP-SAT 周期排程模型，以访问间隔均匀化为目标，结合每日运力、频次要求、14 天最小访问间隔（基于食物支撑周期）等约束，选用 CP-SAT 算法高效求解，输出可落地的每日访问计划。。

4.2 Model Building

在考虑双目标优先性时，我们选择将公平性放于首位。相关研究表明，在食品银行的网络运作中，社会应当优先考虑通过资源分配来消除贫困，而非单纯追求组织能力的扩张\cite{firouz2021equityefficiencytradeofffoodbanknetwork}。若将有效性置于公平之前，低需求、边远站点的弱势群体将被边缘化，违背FBST的组织使命；以公平为前提，先保障基本需求再优化资源配置，方能实现社会价值与运营效率的统一。

4.2.2 Dual-Fairness Guided Frequency Allocation双重公平导向的频次分配

为科学确定各站点年访问频次$k_{i_e}$，本研究构建了以覆盖公平与需求适配公平为核心的双重公平分配机制。其中覆盖公平是保障所有站点基础服务的底线公平，需求适配公平是依据站点需求差异配置额外资源的精准公平。

本机制将覆盖公平（Coverage Equity）作为优先顺位。这一优先级设定的核心逻辑的是：覆盖公平解决的是服务对象是否能获得服务的生存权层面问题，是实现需求适配公平的前提与基础。研究表明，仅以效率为导向的分配算法往往会诱发显著的区域服务差距，且这种不平等会随资源投入的增加而进一步放大 \cite{tang2025contextualbudgetbanditfood}。特别是在食品救助领域，过度追求匹配效率会系统性地边缘化低需求或边远社区；必须通过针对性的覆盖公平机制来纠正地理偏差，保障弱势群体的服务可及性，方能践行FBST项目的“普惠性”核心宗旨\cite{natanzi2026fairshareauditablegeographicfairness}。

覆盖公平（Coverage Equity）：首先设定保障访问次数$k_{base}$，即每个站点每年至少被访问的次数。作为决策变量，参考2019年的平均访问频次（$722\div70\approx10.3$次），将$k_{base}$的取值范围设定为[5,10]。接着计算计算剩余可支配次数$N_{free}$，根据题意可知2021年的总访问次数$N_{total}=730$（365*2），则扣除保障次数后的可支配访问次数为：
N_{free}=N_{total}-70k_{base}
需求适配公平（Demand-Adaptive Equity）：在这一阶段，将个各站点的历史总需求占比作为权重来分配$N_{free}$。站点$i$的历史总需求为$n_i d_i$，即2019年访问次数×平均单次需求量，总历史需求为$\sum_{i=1}^{70} n_i d_i$，因此站点i的年访问次数$k_{i_{e}}$为：
k_{i_{e}} = k_{base} + \text{round}\left(N_{free} \times \frac{n_{i}d_{i}}{\sum_{i=1}^{70} n_{i}d_{i}}\right)

该公式既保障了所有站点的基础服务，又让需求占比高的站点获得更多访问机会，平衡了覆盖公平与需求适配公平。

4.2.3 有效性量化评分及保障频次的影响分析

本文以供需精准匹配为有效性核心判定标准，构建有效性量化评分模型，并基于该模型分析基础保障访问频次kbase对有效性的作用规律，具体如下：

有效性量化评分

step1:计算年有效供应量

考虑卡车单次运输力上限约束，站点$i$的实际年有效供应量为： annual\_eff_{i} = k_{i_{e}} \times \min(d_{i}, d_{0}) 其中，d0=250为单辆卡车单次最大服务家庭数，$d_{i}$为站点i的单次访问需求量的平均值

step2:设置供需失衡惩罚指标

未满足率（惩罚供应不足）： unmet_{i} = \frac{\max(0, D_{i} - annual\_eff_{i})}{D_{i}}
浪费率（惩罚供应过剩）： waste_{i} = \frac{\max(0, k_{i_e} \times d_{0} - D_{i})}{k_{i_e} \times d_{0}}

其中$Di$为站点i的年总需求量

step3:统计综合有效性得分

结合供需匹配度与失衡惩罚，构建站点有效性得分公式：

score_{i} = \frac{annual\_eff_i}{D_{i}} - \alpha \cdot unmet_{i} - \beta \cdot waste_{i}

基于该项目普惠性目标，供应不足的社会危害显著高于资源浪费，因此将未满足率惩罚权重α设为 0.5、浪费率惩罚权重β设为 0.2，契合项目核心诉求。

保障频次对有效性的影响规律

通过对比不同kbase取值下的系统有效性表现，可得出以下核心规律：

随着基础保障访问频次$k_{base}$的提升，站点平均有效性得分呈上升趋势：基础保障机制强化了对低需求站点的资源倾斜，有效降低了此类站点的供应不足风险；同时，由于模型对供应不足的惩罚权重更高，低需求站点得分的提升直接带动了整体平均得分的增长。
总有效供应量则呈现先上升后下降的倒 U 型变化趋势：当kbase处于合理区间时，基础保障既兜底了低需求站点的基本服务，又为高需求站点预留了充足的可支配运力，总有效供应量稳步增长；当kbase过高时，大量运力被强制分配给需求较小的站点，挤占了高需求站点的资源空间，导致高需求站点的供应缺口扩大，最终引发系统总有效供应量下降。

4.2.5 基于基尼系数的站点间服务公平性二次校验

由上述分析结果可见，公益项目的公平原则不仅体现为资源分配的整体公平，更需兼顾站点间服务有效性的均衡公平。为精准量化 70 个站点间的有效性得分差异，本文引入基尼系数作为公平性评价指标，计算公式如下：


G=\frac{\sum_{i=1}^{70} \sum_{j=1}^{70}}{2\times70\times\bar{k_{i}}}

基尼系数G的取值范围为[0,1]，越接近 0 表示站点间服务效能越均衡。因此设定公平性判定阈值：当G<0.2时，认为各站点的服务有效性达到高度均衡状态，选取此时的$k_{base}$作为保障访问的最终次数，并得到各站点的最终年访问次数如下图所示

4.3 Model Solution

确定各站点最优访问频次后，需将年度频次分配至 2021 年 365 天的运营周期中，解决何时访问站点的落地执行问题。本部分构建基于 CP-SAT 的周期排程模型，通过明确约束条件、设计目标函数与选择高效求解算法，实现兼顾运营可行性与服务均衡性的访问计划优化。

4.3.1 Periodic Scheduling Model 周期排程模型

经分析，本模型实质为单目标组合优化问题。模型核心目标是使各站点的年度访问需求在 365 天内尽可能均匀分布，提升服务连续性与客户体验。具体构建过程如下：

决策变量

1.第$i$个站点第$m$次运输的时间$s_{i, m}$，

2.第$i$个站点第$t$天是否被访问
```
a_{i,t}=\left\{\begin{matrix}
1&t\in S_{i}\\
0&t\notin S_{i} 
\end{matrix}\right.
```
目标函数以所有站点实际访问间隔与理想访问间隔的绝对偏差总和最小为目标，量化访问时间分布的均匀性，目标函数如下：
\min Z = \sum_{i=1}^{70} \sum_{m=1}^{k_i - 1} \left| (s_{i, m+1} - s_{i, m}) - T_{i} \right|
约束条件为保证模型解的可行性与合理性，结合项目运营实际设定以下约束：

1.每日的访问总次数不得超过最大运输次数
```
\sum_{i}a_{i,t}\le2
```
2.所有站点必须达到规定访问次数
```
\sum_{t}a_{i,t}=k_{i}
```
3.相邻两次访问不得小于默认间隔
```
s_{i, m+1} - s_{i, m}\ge t_{0}
```

结合实际运营数据验证，当卡车运载食物全部分配给200户家庭时，每户可获得75磅食物，足以支撑14天的需求，因此设定同一站点相邻两次访问的最小间隔 t_0=14 天。

4.3.2 基于CP-SAT的模型求解：

本文构建的模型为单目标组合优化模型，核心聚焦站点访问排班问题。传统混合整数线性规划（MILP）求解大规模排班问题时，存在试错成本高、解空间搜索效率低的局限；而 CP-SAT 算法依托懒惰子句学习机制，可大幅压缩无效解空间，且其自带的冻结功能能在全局优化基础上实现局部精细化调整，进一步提升解的质量，适配大规模排班场景需求，故本文采用该算法求解，所得结果如下。

符号	含义与说明	学术性英文术语
基础参数
`i`	70个常规服务站点的索引 (`i=1, 2, \dots, 70`) [1]	Site Index
`n_i`	站点 `i` 在2019年的历史访问次数 [1]	Historical Visit Frequency
`d_i`	站点 `i` 的历史单次平均需求量（服务客户数） [1]	Average Demand per Visit
`D_i`	站点 `i` 的年度总需求量 (`D_i = n_i \times d_i`)	Annual Aggregate Demand
`d_0`	单辆卡车单次最大服务能力（250户家庭）	Maximum Vehicle Capacity
`N_{total}`	2021年全年的总计划访问次数（365天×2次/天=730次） [1]	Total Annual Service Capacity
`t_0`	同一站点两次访问间的最小安全间隔（14天，基于食物支撑周期）	Minimum Revisit Interval
`\alpha, \beta`	供应不足与资源浪费的惩罚权重系数	Penalty Weighting Coefficients
决策变量
`k_{base}`	每个站点每年至少获得的保障性底线访问次数	Baseline Service Guarantee / Minimum Frequency
`k_{i_e}`	分配给站点 `i` 的最终年度访问频次	Allocated Annual Visit Frequency
`s_{i, m}`	站点 `i` 第 `m` 次被访问的具体日期（天数）	Scheduled Service Date
`a_{i, t}`	0-1 决策变量：站点 `i` 在第 `t` 天是否被访问	Binary Assignment Variable
性能与公平指标
`N_{free}`	扣除基础保障后的剩余可支配访问额度	Residual Allocation Capacity
`annual\_eff_i`	站点 `i` 的年度有效食品供应总量	Total Annual Effective Supply
`unmet_i`	站点 `i` 的需求未满足率（缺货惩罚项）	Unmet Demand Rate (Shortage Penalty)
`waste_i`	站点 `i` 的资源浪费率（过剩惩罚项）	Resource Wastage Rate (Surplus Penalty)
`score_i`	衡量站点服务质量的综合有效性评分	Effectiveness Utility Score
`G`	衡量不同站点间服务效能均衡程度的基尼系数	Gini Coefficient (Equity Metric)
`T_i`	站点 `i` 的理想均匀访问间隔周期 (`365 / k_{i_e}`)	Ideal Recurrence Interval

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