Modern experimental designs often face the so-called treatment cardinality constraint, which is the constraint on the number of included factors in each treatment. Experiments with such constraints are commonly encountered in engineering simulation, AI system tuning, and large-scale system verification. This calls for the development of adequate designs to enable statistical efficiency for modeling and analysis within feasible constraints. In this work, we study two-level designs under this $k$-treatment cardinality constraint (TCARD), where the design matrix $\mathbf{X} \in \{0,1\}^{n \times p}$ has constant row sums equal to $k$. Although TCARDs are closely related to balanced incomplete block designs (BIBDs), exact BIBD structure is unavailable for many practical $(n,p,k)$ combinations. This leads to the notion of nearly balanced TCARDs, which we prove minimize the first two components of the generalized word-length pattern. We also show that good projection behavior in this setting is governed by two count-based regularities: balanced factor replications and uniform pairwise concurrences. Motivated by this characterization, we then propose the Balanced Concurrence Deviation ($Φ_{\mathrm{BCD}}$), a model-free objective that jointly penalizes replication imbalance and concurrence dispersion. We further show that this criterion is closely connected to classical optimality principles, including $(M,S)$-optimality, centered $\mathrm{UE}(s^2)$ criterion, and Bayesian $D$-optimality. To construct designs minimizing $Φ_{\mathrm{BCD}}$, we develop a coordinate-exchange (CE) algorithm with efficient incremental updates, together with a simulation-based procedure for calibrating the criterion weights to the intended downstream task. Numerical experiments confirm that the proposed method compares favorably with existing alternatives across a range of problem sizes and constraint strengths.

翻译：现代实验设计常面临所谓的“治疗基数约束”，即每个处理中涵盖因子数量的限制。此类约束在工程仿真、人工智能系统调优及大规模系统验证中普遍存在，这要求开发能够满足可行约束并实现建模与分析统计效率的充分设计。本研究针对$k$-治疗基数约束（TCARD）下的两水平设计展开探讨，其中设计矩阵$\mathbf{X} \in \{0,1\}^{n \times p}$的每行元素之和恒等于$k$。尽管TCARD与平衡不完全区组设计密切相关，但许多实际$(n,p,k)$组合无法直接采用精确BIBD结构，由此引出近似平衡TCARD的概念。我们证明此类设计可最小化广义字长模式的前两个分量，并揭示在此设定下，良好投影特性取决于两种基于计数的正则性：平衡因子重复与均匀配对共现。基于这一特征化结果，我们提出无模型优化目标——平衡偏差度量（$Φ_{\mathrm{BCD}}$），该准则联合惩罚重复不平衡与共现分散性。进一步证明该准则与$(M,S)$最优性、中心化$\mathrm{UE}(s^2)$准则及贝叶斯$D$最优性等经典优化原则存在紧密关联。为构建使$Φ_{\mathrm{BCD}}$最小化的设计，我们开发了带高效增量更新的坐标交换算法，并设计基于模拟的标定流程以适配目标下游任务的准则权重。数值实验表明，在多种问题规模与约束强度下，所提方法相较现有替代方案具有显著优势。