Стохастична мережево-часова модель PERT/CPM З імовірнісними переробками для оцінювання строків ІТ-проєкті

UKR: У статті розглянуто задачу імовірнісного оцінювання строків завершення ІТ-проєктів за умов невизначеності тривалостей робіт і повторних циклів доопрацювання результатів після контрольних перевірок (rework). Запропоновано стохастичну мережево-часову модель на основі PERT/CPM, у якій базові тривалості робіт задаються триточковими оцінками PERT, а механізм переробок формалізовано через ймовірність непроходження контрольного етапу та випадкову тривалість переробки (з можливістю врахування ефекту навчання). Для отримання розподілу часу завершення проєкту застосовано імітаційне моделювання методом Монте-Карло з мережевим розрахунком ранніх термінів старту/завершення. Обчислюються перцентильні строки завершення (P50/P80/P90), ймовірність виконання заданого дедлайну та показники критичності робіт і чутливості параметрів (tornado-аналіз). Запропонований підхід забезпечує більш адекватне оцінювання календарних ризиків порівняно з детермінованими планами та може бути використаний для обґрунтування буферів і пріоритетів управління якістю й ризиками в ІТ-проєктах.

ENG: Information technology (IT) projects are often affected by two sources of uncertainty. First, the duration of each activity is not known in advance and can vary widely due to technical complexity, changing requirements, and team-related factors. Second, many deliverables must pass through multiple verification and acceptance gates—such as code review, quality assurance (QA) testing, security checks, and user acceptance testing (UAT). At these gates, outcomes may fail to meet agreed criteria and therefore require rework. This iterative “do–check–rework” cycle is a common reason why projects deviate from deterministic schedules. Traditional CPM schedules and many classical PERT applications typically treat activities as one-pass tasks and do not explicitly represent repeated rework loops. As a result, they can underestimate completion-time risk, especially in the upper tail of the distribution, and provide overly optimistic target dates. This paper presents a stochastic network–time scheduling model that integrates the PERT/CPM framework with an explicit probabilistic representation of rework. The project is modeled as a precedence network where nodes represent activities and directed links represent finish-to-start constraints. Each activity has a base duration described by a three-point estimate (optimistic, most likely, and pessimistic) and is treated as a random variable following a PERT-type distribution. To represent rework, each activity may be associated with one or more verification gates. For each gate, the model specifies a probability that the activity fails the gate and must be reworked, and a probability distribution for the rework effort (also described using three-point estimates). The number of rework cycles is modeled as a random count driven by repeated attempts until the gate is passed. Optionally, the model includes a learning effect: repeated rework cycles may become faster over time as the team gains understanding, defects become localized, and corrective actions become more targeted. To obtain project completion-time forecasts, the study uses Monte Carlo simulation. In each simulation run, base durations and rework outcomes are sampled for all activities and gates, and then a standard CPM forward pass computes start and finish times consistent with network dependencies. Repeating this process many times produces an empirical distribution of total project duration. From this distribution, the paper reports percentile completion dates (such as median, 80th percentile, and 90th percentile) and the probability of meeting a specified deadline. Percentile-based dates are emphasized because they are more suitable for reliable commitments than mean-based estimates in the presence of rework-driven tail risk. In addition to overall completion-time forecasts, the paper provides diagnostic analyses to support risk management. First, an S-curve (empirical cumulative distribution) visualizes the completion-time distribution and allows direct reading of deadline probabilities and percentile dates. Second, parameter sweeps demonstrate how deadline success probability changes when rework probability increases at a specific gate and how upper-percentile completion times respond to changes in the learning effect. Third, a tornado-style one-at-a-time sensitivity analysis ranks which rework-related parameters have the greatest impact on a risk-oriented completion metric (e.g., the 80th percentile). Finally, the paper computes a Criticality Index for each activity—the fraction of simulation runs in which the activity lies on the critical path—showing that criticality can shift between branches under uncertainty and highlighting a “critical core” of activities that most frequently drive the project finish date. The results indicate that explicitly modeling probabilistic rework materially affects forecasted completion dates and expands the uncertainty range, particularly in the upper tail. The approach supports more realistic buffer sizing and deadline-risk assessment than deterministic planning. Sensitivity and criticality outputs provide actionable guidance by identifying which gates (e.g., QA or UAT) and which activities should be prioritized for quality improvements, automation, clearer acceptance criteria, or process changes that reduce the frequency and cost of rework. Overall, the proposed stochastic PERT/CPM model with probabilistic rework offers a practical and mathematically grounded method for forecasting completion dates and quantifying deadline risk in IT projects with multi-stage verification and iterative rework. Future research may extend the model to incorporate resource constraints, correlations between activities and risks, and data-driven estimation of model parameters from project repositories and quality metrics.