Nowadays the advent of new types of projects such as startups, maintenance, and education make a revolution in project management, so that, classical project scheduling methods are incapable in analyzing of these stochastic projects. This study considers a time–cost trade-off project scheduling problem, where the structure of the project is uncertain. To deal with the uncertainties, we implemented Graphical Evaluation and Review Technique (GERT). The main aim of the study is to balance time and the amount of a non-renewable resource allocated to each activity considering the finite-time horizon and resource limitations. To preserve the generality of the model, we considered both discrete and continuous distribution functions for the activity’s duration. From a methodological standpoint, we proposed an analytical approach based on the Markov Decision Process (MDP) and Semi-Markov Decision Process (SMDP) to find the probability distribution of project makespan. These models are solved using the value iteration and a finite-horizon Linear Programming (LP) model. Two randomly generated examples explain the value iteration for models in detail. Furthermore, seven example groups each with five instances are adopted from a well-known data set, PSPLIB, to validate the efficiency of the proposed models in contrast to the two extensively-studied methods, Genetic algorithm (GA) and Monte-Carlo simulation. The convergence of the GA and simulation results to those of MDP and SMDP represent the efficiency of the proposed models. Besides, conducting a sensitivity analysis on the project completion probability with respect to the available resource, gives a good insight to managers to plan their resources.

如今，启动、维护、教育等新型项目的出现给项目管理带来了一场革命，传统的项目调度方法已经无法分析这些随机项目。本研究考虑时间成本权衡项目调度问题，其中项目结构不确定。为了应对不确定性，我们实施了图形评估和审查技术（GERT）。该研究的主要目的是考虑到有限时间范围和资源限制，平衡分配给每项活动的时间和不可再生资源的数量。为了保持模型的通用性，我们考虑了活动持续时间的离散和连续分布函数。从方法论的角度来看，我们提出了一种基于马尔可夫决策过程（MDP）和半马尔可夫决策过程（SMDP）的分析方法来寻找项目完工时间的概率分布。这些模型使用值迭代和有限范围线性规划 (LP) 模型进行求解。两个随机生成的示例详细解释了模型的值迭代。此外，从著名的数据集 PSPLIB 中采用了七个示例组，每个示例组有五个实例，以验证所提出的模型与两种广泛研究的方法（遗传算法（GA）和蒙特卡洛模拟）的效率。 GA 和仿真结果与 MDP 和 SMDP 的收敛性代表了所提出模型的效率。此外，对可用资源的项目完成概率进行敏感性分析，可以为管理者规划资源提供良好的洞察力。