Abstract: Project scheduling in a deterministic environment has been explored by a number of scholars for decades. However, in the existing dramatically changing market environment, complex engineering projects (e.g., “The Belt and Road Initiative” transnational projects, PPP projects, etc.) are facing increasingly severe risks and uncertainties, such as inaccurate estimation of activity duration, untimely supply of resources, machine breakdown, severe weather conditions, and changes in design. These uncontrollable factors can disrupt the orderly execution of project activities and cause confusion in organization and coordination, leading to project delays and cost overruns. Therefore, it is necessary to present efficient approaches to these uncertainties.
Robust project scheduling is to develop a schedule with strong anti-interference ability and flexible reactive strategies by taking into account the disturbance of uncertainties, having become a crucial research topic in recent decades. Existing robust scheduling studies have assumed that all activities can only be constructed in a fixed logical sequence. In practical engineering, however, the sequence of logical construction between many activities can be changed, which is known as soft logic. The research on soft logic mostly focuses on project duration and cost optimization, and there are no studies known to authors introducing soft logic to project scheduling in an uncertain environment. In this context, this paper studies robust project scheduling optimization considering soft logic, which has strong theoretical value and practical significance.
Firstly, we analyze the impact of soft logic on project scheduling, including project duration, solution robustness, and the number of subsequent activities, which provides the theoretical basis for the proposed optimization model in this paper.Soft logic can make project scheduling more flexible. On the one hand, by reasonably adjusting the construction sequence of activities, the critical path of the schedule can be changed, thus affecting project duration. On the other hand, the precedence relationship among activities can be also regulated in the same manner, which in turn has an impact on the free float of activities and ultimately affects the robustness of project. We also quantify the specific impact of three types of soft logic on the number of subsequent activities, which is widely considered in the development of robustness measures.
Secondly, an optimization model is formulated and solved by the e-constraint algorithm. We improve the robustness of the project schedule by inserting time buffer in the project activities, which extends project duration. Therefore, the objective of the model is to achieve the trade-off optimization between minimized project duration and maximized robustness, taking into account the soft logic relationships between activities and project deadline. The robustness objective function uses a measure based on free float and accounts for the magnitude of delay risk, allowing time buffer to be prioritized for allocation to activities with higher delay risk and more subsequent activities. Furthermore, an e-constraint algorithm is specifically developed to identify Pareto optimal solutions to the studied problem. We maximize the schedule robustness with the objective of minimizing the project duration as a constraint. In doing so, the proposed model is transformed into multiple single-objective optimization models, which helps to identify all Pareto solutions in a fairly small number of steps.
Finally, the effectiveness and superiority of the model and algorithm for robust scheduling considering soft logic are verified through a typical practical case and Monte Carlo simulation. We respectively solve the duration-robust scheduling optimization problem under the assumptions of soft logic and traditional fixed logic. The results show that soft logic is more powerful than fixed logic, providing shorter project duration and stronger robustness. We also suggest that there are two approaches for soft logic strategies to improving robustness without sacrificing project duration. One is to reduce the time buffer formed by the network through soft logic while leaving space for the addition of artificial buffers, and the other is to increase the time buffer formed by the network through soft logic. Furthermore, Monte Carlo simulation results imply that compared with fixed logic, project schedule considering soft logic is more stable in the execution of its activities and on-time completion and is less vulnerable to disruptions caused by various uncertainties, and that with the increase in the level of uncertainty, soft logic is increasingly superior to fixed logic.
The research results of this article can provide quantitative support for project managers' decisions to make robust schedules considering soft logic in uncertain environment. Owing to the practical limitations imposed by resource availability, the flexibility afforded by soft logic may be limited. Therefore, a future research direction would be robust resource-constrained project scheduling with soft logic.

Key words: robust scheduling, soft logic, bi-objective programming, e-constraint algorithm

摘要： 不确定环境下,基准进度计划的稳定性对项目的顺利实施十分重要,而软逻辑能够在项目遇到扰动时提供更强的自由度和调节能力。本文研究考虑软逻辑的鲁棒项目调度优化问题。首先,分析三种类型的软逻辑对项目工期和解鲁棒性的影响;然后,构建基于软逻辑的双目标鲁棒项目调度优化模型,并设计e-约束算法对其进行求解;最后,通过一个典型的实际案例和蒙特卡洛模拟验证了模型和算法的可行性与有效性,结果表明考虑软逻辑可以使项目工期和解鲁棒性同时得到优化。本文将软逻辑应用于鲁棒项目调度优化问题,能够进一步提高项目调度方案应对不确定性因素干扰的能力。

关键词: 鲁棒性调度, 软逻辑, 双目标, e-约束算法