## 1 Introduction  
In order to meet the ever-increasing demand for mineral resources consumption and supply, increasing mining production capacity is an important consideration for designers and managers of nonferrous underground mines, however, adequate safety and reasonable mining operation costs must be guaranteed. Currently, long-hole stage open stoping with subsequent backfill is the most efficient mining method and represents the development direction of large-scale underground mining in nonferrous metal mines [1; 2; 3]. In this mining method, a traditional bottom-up mining sequence is usually adopted, in which the ore body is vertically divided into a number of mining stages, as illustrated in Figure (a)a, where the stopes (1, 3, 5) in Stage-1 are excavated and then backfilled to provide a base platform for the mining of upper stopes (2, 4, 6) in Stage-2. Meanwhile, continuous stopes are divided along the horizontal direction of each stage, and the primary stopes (1, 2) are excavated preferentially and backfilled with cement to serve as artificial vertical pillars for the following excavations of secondary stopes (3, 4, 5, 6), which are mostly backfilled without adding cement to save costs. Undoubtedly, due to the higher efficiency of long-hole blasting, larger stope sizes can increase the stope production capacity and reduce the mining costs per excavated ore. But due to a larger exposed surface of the surrounding rockmass and backfill, it usually leads to a higher safety risk of stope failure. Therefore, the determination of reasonable dimension parameters (height, length and width) of the stopes is significant to the design of the mining process considering the stopping stability, capacity and costs. To further increase the stope production capacity and reduce the cost of mined ore, the double stage open stoping for primary stopes (shown as Stope 1\({}^{\prime}\) in Figure 1b) in some mines with high-dipping thick stable ore bodies can be employed to enlarge the advantages of mining efficiency and cost-saving [4,5]. The combination of vertically adjacent primary stopes (1, 2) saves construction costs to develop the entry drifts and ore drawing structures, and eliminates the waiting time of backfilling the lower stope void to build the platform for the upper stope. However, the combination of excavated primary stopes in the vertical direction would result in doubling the height of the primary stopes, thereby reducing the stability of side-exposed rock walls. This presents an even greater challenge in balancing stope stability and dimensions. Therefore, it can be seen that the benefits and safety management of stage open stoping mining is highly dependent on the optimization of stope dimensions and the corresponding indexes of mining capacity and costs, which is a hot topic in the field of underground mining with backfill. Empirical charts, analytical models and numerical simulations are usually used to achieve the optimization and reasonable balance of stability and dimensions of open stopes, and the following fruitful research findings and typical applications are obtained. The Mathews stability graph [6] is a world-wide method used to build up an empirical chart between the stability index and the shape factor with different dimension parameters of exposed side walls and roof around stope voids. Subsequently, [PERSON] [7] expanded the Mathews database and proposed an extended stability graph method based on logistic regression analysis which can calculate the failure probability of a stope void. [PERSON] et al. [8] further updated the extended Mathews stability graph by fitting a formula that can directly calculate the stope shape factors corresponding to a 95% stability probability for a specific underground mining project. Considering the effects of exposure time on the inclined stope stability in different sizes, [PERSON] et al. [9] proposed a modified stability synthetic graph for stope dimension selection.