Abstract: Resources-constrained project scheduling problems widely exist in construction engineering, equipment manufacturing and other enterprises, and have significant practical value. Most of the classical projects scheduling problems are deterministic, which assumes that the internal and external environment will not change. However, in real life, there are a number of uncertainties in most projects. In an uncertain environment, a schedule made during the project design is likely to be delayed due to interference. It is particularly important to set up a scheduling plan with strong anti-interference ability in a complex and uncertain environment. Time and cost are two important indicators in project scheduling, and the robustness of the schedule is the key to ensuring the smooth implementation of the project in uncertainty. It is particularly important to set a scheduling plan with a strong anti-interference ability in the complex uncertain environment.
In this paper, we study a multi-objective project scheduling optimization problem in an uncertain environment. We try to balance the completion time, robustness and delay penalty cost to find a solution to meet various needs. What is more, we first define the problem and list the corresponding symbols and expressions of calculation formulas. This paper proposes the necessity of setting resources buffer on the basis of time buffer. The resources buffer can offset the resources conflict in an original project schedule caused by the delay of a certain process for some reason or the unavailability of part of the planned available resources when an activity is executed, so as to ensure that the overall operation of the project is not disturbed. The time buffer and resources buffer are combined, and the effectiveness of the combination is further illustrated by an example, which is further verified in the data simulation. Then, a multi-objective robust optimization model of this problem is constructed and introduced in detail. A NSGA-II algorithm is a multi-objective genetic algorithm based on non-dominated sorting, which has been widely studied and applied in solving multi-objective problems, so it is chosen to solve the problem. In the fourth part of this paper, an improved multi-objective non-dominated sorting genetic algorithm is designed, and the relevant steps of the algorithm are improved to make it more suitable for the problems proposed in this paper. The improved steps are described in detail in the article. The algorithm designed in this paper shortens the time required to solve the problem by ensuring the feasibility of the solution, and adds an uncertain environment to make the solution closer to reality, and also helps the algorithm to screen out better individuals through the simulation environment so as to enter the next iteration. Finally, numerical experiments are designed in the fifth part of the paper. In order to reflect the performance of the improved algorithm in this paper, the traditional non-dominated sorting genetic algorithm is compared with the improved algorithm. Through the generated standard sample set, the control experiments under different activity numbers and different duration constraints are designed. The output of the data experiment is presented in the attached table. The effectiveness and feasibility of the algorithm are verified by a large number of experiments, and the Pareto optimal solution obtained by the algorithm is tested in the uncertain environment, and the test results further verify the performance of the obtained solution. For the output optimal solution set, the manager can choose the appropriate solution according to personal preferences.
Finally, according to the mathematical model and data experiment designed in this paper, the following conclusions are drawn: (1)Having a resource buffer can effectively deal with the problem of resource usage conflicts caused by activity delays. With the help of the optimization model proposed in this paper, the resources buffer can be reasonably allocated among the activities, and then the robustness of the project schedule can be effectively improved. (2)By improving the NSGA-II algorithm, it is not difficult to find that its performance is better than the former, and it is more suitable for solving this problem. (3)The robustness of the project schedule increases with the extension of the construction period. The results can provide a reference for project managers to weigh objectives and make progress plans in an uncertain environment. It should be pointed out that the research in this article does not consider the cost of adding buffer, which needs to be further discussed in the next study.

Key words: project scheduling, robustness, multi-objective optimization, improved NSGA-II

摘要： 在复杂不确定性环境下设定一个具有较强抗干扰能力的调度计划尤为重要。本文研究不确定性环境下的多目标项目调度优化问题,以最小化完工时间、最大化鲁棒性、最小化延迟惩罚成本为目标,通过权衡三者寻找满足多方面需求的解。文章首先对所研究的问题进行了一个界定说明,将时间缓冲与资源缓冲相结合,并通过一个示例进一步说明将时间缓冲与资源缓冲结合的有效性。随后构建问题的多目标优化模型,设计改进的多目标非支配排序遗传算法 (Non-dominated Sorting Genetic Algorithm II,NSGA-II),并加入不确定性环境使其所得的解更贴近现实。最后在随机生成的标准算例集合上对算法进行测试,验证了该算法的可行性和有效性,并在不确定性环境下测试了算法所得到的Pareto最优解。研究结果可为项目管理者在不确定性环境下权衡目标、制定进度计划提供参考。

关键词: 项目调度, 鲁棒性, 多目标优化, 改进的NSGA-II