The hydropower system in the Upper Yellow River (UYR), one of the largest hydropower bases in China, plays a vital role in the energy structure of the Qinghai Power Grid. Due to management difficulties, there is still considerable room for improvement in the joint operation of this system. This paper presents a general LINGO-based integrated framework to study the operation of the UYR hydropower system. The framework is easy to use for operators with little experience in mathematical modeling, takes full advantage of LINGO's capabilities (such as its solving capacity and multi-threading ability), and packs its three layers (the user layer, the coordination layer, and the base layer) together into an integrated solution that is robust and efficient and represents an effective tool for data/scenario management and analysis. The framework is general and can be easily transferred to other hydropower systems with minimal effort, and it can be extended as the base layer is enriched. The multi-objective model that represents the trade-off between power quantity (i.e., maximum energy production) and power reliability (i.e., firm output) of hydropower operation has been formulated. With equivalent transformations, the optimization problem can be solved by the nonlinear programming (NLP) solvers embedded in the LINGO software, such as the General Solver, the Multi-start Solver, and the Global Solver. Both simulation and optimization are performed to verify the model's accuracy and to evaluate the operation of the UYR hydropower system. A total of 13 hydropower plants currently in operation are involved, including two pivotal storage reservoirs on the Yellow River, which are the Longyangxia Reservoir and the Liujiaxia Reservoir. Historical hydrological data from multiple years (2000-2010) are provided as input to the model for analysis. The results are as follows. 1) Assuming that the reservoirs are all in operation (in fact, some reservoirs were not operational or did not collect all of the relevant data during the study period), the energy production is estimated as 267.7, 357.5, and 358.3×108 KWh for the Qinghai Power Grid during dry, normal, and wet years, respectively. 2) Assuming that the hydropower system is operated jointly, the firm output can reach 3110 MW (reliability of 100%) and 3510 MW (reliability of 90%). Moreover, a decrease in energy production from the Longyangxia Reservoir can bring about a very large increase in firm output from the hydropower system. 3) The maximum energy production can reach 297.7, 363.9, and 411.4×108 KWh during dry, normal, and wet years, respectively. The trade-off curve between maximum energy production and firm output is also provided for reference.