Bipartite graphs are widely used to model relationships among diverse entities in domains such as gene co-expression networks, collaboration networks, and customer-product interactions. A fundamental problem in analyzing bipartite graphs is the maximum balanced biclique (MBBC) search, which identifies the maximum fully connected subgraph with an equal number of vertices on both sides in the given bipartite graph. Despite its utility, the MBBC model suffers from practical limitations: its strict all-to-all connectivity and exact size-equality requirements make it impractical for noisy, incomplete real-world bipartite data. To overcome these limitations, we propose the maximum balanced k-biplex (MBKBP) model, which relaxes the stringent requirements of MBBC. In MBKBP, each vertex is allowed to miss up to k neighbors on the opposite side of the bipartite graph, and a user-defined parameter$\delta$ensures approximate balance between the two vertex sets. This flexibility enhances robustness to noise, accommodates incomplete data, and broadens the model's applicability. To compute the MBKBP in a given bipartite graph, a baseline approach involves enumerating all maximal balanced k-biplexes and identifying the largest one. However, as confirmed by our experiments, this approach is computationally inefficient. To address this challenge, we introduce the concept of$(z_{L},\ z_{R})$search space and propose a new framework to compute the MBKBP. By generating a series of smaller$(z_{L)}z_{R})$search spaces, our framework significantly reduces the number of maximal k-biplexes that need to be explored. Additionally, we leverage the$\delta$-balance property to refine the search spaces further and develop three categories of pruning rules to minimize computational overhead. Extensive experiments on real-world bipartite graphs demonstrate that our algorithm achieves up to three orders of magnitude speedup compared to baseline approache, showcasing its efficiency and practicality for bipartite graph analysis.