The project scheduling problem is essential both in the theoretical part, as in the field of operational research, and practice, with the project management in corporate environments. Integer linear programming formulations indexed on time are studied for the Resource-Constrained Project Scheduling Problem (RCPSP). Moreover, the multi-skill, multiple modes, and time lag constraints are taken into consideration. The objective of the RCPSP is to minimize the makespan. The formulations are solved with the default branch-and-cut algorithm of the solver Gurobi Optimizer. The formulations and solver are analyzed concerning the runtime, the number of optimal solutions, and the gap on the resolution of more than 2000 instances. Results indicate the solver can have better performance when instances with up to 50 activities are solved. Then, to develop models to handle hard instances of this problem is a challenge. Moreover, it can bring significant advantages to the corporate environment, helping managers to make accurate decisions and reduce costs.

中文翻译：

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考虑多技能、多模式以及最小和最大时间滞后的 RCPSP 整数线性规划公式


项目调度问题在理论部分（如运筹学领域）和实践（如企业环境中的项目管理）中都是至关重要的。研究了针对资源受限项目调度问题（RCPSP）的以时间为索引的整数线性规划公式。此外，还考虑了多技能、多模式和时滞约束。 RCPSP 的目标是最大限度地缩短完工时间。使用求解器 Gurobi Optimizer 的默认分支剪切算法求解公式。对公式和求解器的运行时间、最优解的数量以及超过 2000 个实例的分辨率差距进行了分析。结果表明，当求解最多 50 个活动的实例时，求解器可以具有更好的性能。那么，开发模型来处理这个问题的困难实例是一个挑战。而且，它可以给企业环境带来显着的优势，帮助管理者做出准确的决策并降低成本。