--- title: "Critical path method (CPM) — 关键路径法 (CPM)" source: "https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)" wiki: "systems-analysis.info/int" article: "Critical_path_method_(CPM)_—_关键路径法_(CPM)" language: "zh" categories: - "Category:Chinese" - "Category:Operations research" - "Category:Project management" revision_id: 1336 wiki_created_at: 2026-09-06T22:47:09Z wiki_modified_at: 2026-09-06T22:47:09Z downloaded_at: 2026-09-07T22:45:31Z --- # Critical path method (CPM) — 关键路径法 (CPM) **关键路径法 (CPM)**（英语：*Critical Path Method*，*CPM*）是一种用于网络计划和项目管理的确定性方法，用于计算工作序列和项目可能的最短完成时间。关键路径是网络模型中持续时间最长的路径；该路径上任何活动的延迟都会推迟整个项目的结束时间^{[\[1\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-enwiki-1)[\[2\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-kelley1959-2)}。 ## 历史 CPM由詹姆斯·凯利（James Kelley，来自Remington Rand）和摩根·沃克（Morgan Walker，来自DuPont）于20世纪50年代末开发。他们于1959年发表的报告开启了该方法的工业应用；后来，作者们发表了一篇关于CPM起源的历史回顾^{[\[2\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-kelley1959-2)[\[3\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-origins-3)}。与此同时，美国开发了概率方法PERT，CPM经常与之结合使用^{[\[1\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-enwiki-1)}。 ## 基本概念 - **活动** (activity) — 具有持续时间 **d** 并依赖于前置活动的一项工作。 - **事件/里程碑** — 没有持续时间的时间点，用于标记项目的状态。 - **关键路径** — 模型中从开始到结束总持续时间最长的一系列活动；该路径上的活动总浮动时间为零^{[\[1\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-enwiki-1)}。 - **前置关系** 通过FS、SS、FF、SF关系以及可能的滞后时间来定义；实践中通常使用PDM / AON（活动节点法），而历史上的ADM / AOA（活动箭线法）则较少使用^{[\[4\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-pdm-4)[\[5\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-adm-5)[\[6\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-dependency-6)}。 ## 日期与浮动时间的计算计算通过网络图的正向和反向推导完成。 **正向推导**（最早日期）：对于起始活动：**ES = 0**（或根据所采用的时间刻度）；对于每个活动 *j*：**ES_j = max{ 所有前置活动 *i* 的 EF_i }**； **EF = ES + d**。 **反向推导**（最晚日期）：对于结束活动：**LF** 等于项目的总工期（或最终里程碑的最早完成时间）；对于每个活动 *j*：**LS = LF − d**；**LF_j = min{ 所有后续活动 *s* 的 LS_s }**^{[\[7\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-baker-7)[\[8\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-cmu-8)}。 **浮动时间** (float/slack)： **总浮动时间 (TF)**: **TF = LS − ES = LF − EF** — 在不影响项目整体完成时间的前提下，一项活动可以延迟的时间。 **自由浮动时间 (FF)**: **FF = min{ ES_succ } − EF** — 在不影响任何后续活动最早开始时间的前提下，一项活动可以延迟的时间^{[\[9\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-pressbooks-9)[\[10\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-pmi-ff-10)}。 ## 关系与滞后时间在PDM中，支持四种基本依赖关系类型：**FS**（完成-开始）、**SS**（开始-开始）、**FF**（完成-完成）、**SF**（开始-完成）。每种关系都可以带有滞后时间（正值或负值），例如 **FS + 2d** 指后续活动在前置活动完成后两天开始^{[\[6\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-dependency-6)}。 ## 进度压缩 (赶工与快速跟进) 项目工期的缩短可以通过以下方式实现： - **赶工 (Crashing)** — 通过增加额外成本来有针对性地缩短关键活动的持续时间；优先选择成本“斜率”最低的活动进行压缩。 **成本斜率**（每单位压缩时间的成本）：**(C_crash − C_normal) / (D_normal − D_crash)**^{[\[11\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-crash-11)}。 - **快速跟进 (Fast-tracking)** — 将原本顺序执行的活动改为并行执行（例如，FS → SS），这会增加返工的风险^{[\[1\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-enwiki-1)}。 ## 应用领域 CPM被用于制定建筑、能源、机械工程、IT项目和研发（R&D）领域的进度计划。该方法是项目管理和系统工程标准（如PMI、NASA）中的基础方法，并得到各种规划软件的广泛支持^{[\[1\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-enwiki-1)[\[12\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-pmi-12)[\[13\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-nasa-13)}。 ## 优点 - 清晰地识别关键路径和“瓶颈”；帮助项目经理确定关注的优先事项^{[\[1\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-enwiki-1)}。 - 计算最早/最晚日期和浮动时间的规则简单；变更带来的影响透明可见^{[\[7\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-baker-7)}。 - 与PDM/甘特图以及用于分析加速方案的工具兼容^{[\[4\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-pdm-4)}。 ## 局限性 - 基本的CPM模型未考虑资源限制和持续时间的随机性；为此，需要应用资源平衡和扩展方法（例如，*关键链*）^{[\[1\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-enwiki-1)}。 - 当使用复杂的依赖关系和滞后时间时，浮动时间的解释可能变得不那么直观；建议在合同文件中对浮动时间的计算规则进行正式定义和控制^{[\[10\]](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_note-pmi-ff-10)}。 ## 与其他方法的关系 - PERT — 采用概率性的持续时间估算（三点估算，β分布近似）；而CPM使用确定性的持续时间。 - 甘特图 — 进度的日历化可视化工具，常与CPM/PERT结合使用。 - 网络计划技术 — 一组方法的总称（包括CPM、PERT、ADM/AOA、PDM/AON）。 ## 参见 - PERT - 甘特图 - 网络计划技术 ## 参考文献 - Kelley, J. E.; Walker, M. R. (1959). *Critical‑Path Planning and Scheduling*. IRE‑AIEE‑ACM '59 (Eastern). *ACM Digital Library*. [14] - Kelley, J. E.; Walker, M. R. (1989). *Origins of CPM: A Personal History*. PM Network. *PMI*. [15] - PMI. *PMBOK® Guide*. [16] - NASA. *Systems Engineering Handbook* (SP‑2016‑6105 Rev2). [17] - «Critical path method». *Wikipedia (en)*. [18] - «Precedence diagram method». *Wikipedia (en)*. [19] - «Dependency (project management)». *Wikipedia (en)*. [20] - «Arrow diagramming method». *Wikipedia (en)*. [21] - Baker, S. L. *Critical Path Method (CPM) — Analysis Steps*. University of South Carolina. [22] - «Creating an Activity Network Diagram». *CSU Pressbooks*. [23] - «Crashing Example». An‑Najah National University (教学示例). [24] ## 注释 1. ↑ ^{[1.0](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-enwiki_1-0)} ^{[1.1](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-enwiki_1-1)} ^{[1.2](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-enwiki_1-2)} ^{[1.3](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-enwiki_1-3)} ^{[1.4](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-enwiki_1-4)} ^{[1.5](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-enwiki_1-5)} ^{[1.6](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-enwiki_1-6)} “Critical path method”. *Wikipedia (en)*. [1] 2. ↑ ^{[2.0](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-kelley1959_2-0)} ^{[2.1](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-kelley1959_2-1)} Kelley, J. E., Jr.; Walker, M. R. (1959). «Critical-Path Planning and Scheduling». *IRE‑AIEE‑ACM '59 (Eastern)*. *ACM Digital Library*. [2] 3. [↑](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-origins_3-0) Kelley, J. E.; Walker, M. R. (1989). «Origins of CPM: A Personal History». *PM Network*. *Project Management Institute*. [3] 4. ↑ ^{[4.0](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-pdm_4-0)} ^{[4.1](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-pdm_4-1)} «Precedence diagram method». *Wikipedia (en)*. [4] 5. [↑](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-adm_5-0) «Arrow diagramming method». *Wikipedia (en)*. [5] 6. ↑ ^{[6.0](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-dependency_6-0)} ^{[6.1](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-dependency_6-1)} «Dependency (project management)». *Wikipedia (en)*. [6] 7. ↑ ^{[7.0](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-baker_7-0)} ^{[7.1](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-baker_7-1)} Baker, S. L. «Critical Path Method (CPM) — Analysis Steps». University of South Carolina. [7] 8. [↑](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-cmu_8-0) «Fundamental Scheduling Procedures». *Project Management*, Carnegie Mellon University. [8] 9. [↑](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-pressbooks_9-0) «Creating an Activity Network Diagram». *Project Management — Navigating the Complexity*. Cleveland State University Pressbooks. [9] 10. ↑ ^{[10.0](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-pmi-ff_10-0)} ^{[10.1](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-pmi-ff_10-1)} «Critical Path Method Calculations». *PMI*. [10] 11. [↑](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-crash_11-0) «Crashing Example». An‑Najah National University (教学示例). [11] 12. [↑](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-pmi_12-0) PMI. *A Guide to the Project Management Body of Knowledge (PMBOK® Guide)*. [12] 13. [↑](https://systems-analysis.info/int/Critical_path_method_(CPM)_%E2%80%94_%E5%85%B3%E9%94%AE%E8%B7%AF%E5%BE%84%E6%B3%95_(CPM)#cite_ref-nasa_13-0) NASA. *Systems Engineering Handbook* (NASA/SP‑2016‑6105 Rev2). [13]