We consider the stochastic production planning problem with product substitution, which can be decomposed into several optimization subproblems with sequential decisions. The decision variables in each time period include (1) the product substitution decision and (2) the recipe input quantity decision. The goal is to minimize the total of production cost, holding cost, and shortage cost, while achieving a service level for demand satisfaction.

Since this optimization model involves analytically intractable probabilistic formulation, traditional mathematical programming techniques cannot be readily applied. We develop the deterministic SPLINE and the stochastic R-SPLINE algorithms for different scenarios. The probability generating function is embedded into the deterministic algorithm to exactly calculate the desired performance measures, which is reasonable when dealing with independent data (with a small number of product classes as well). The stochastic R-SPLINE algorithm uses simulation to estimate the desired probabilistic measures, allowing correlations between different production recipes as well as between different demand classes. We also present a convergence analysis for the stochastic R-SPLINE algorithm. Experimental results demonstrate the efficiency of the developed algorithms compared to other existing approaches.

我们考虑产品替代的随机生产计划问题，它可以分解为几个具有顺序决策的优化子问题。每个时间段的决策变量包括（1）产品替代决策和（2）配方投入量决策。目标是最小化生产成本、持有成本和短缺成本的总和，同时达到满足需求的服务水平。

由于该优化模型涉及分析上难以处理的概率公式，因此传统的数学规划技术不能轻易应用。我们针对不同场景开发了确定性SPLINE 和随机 R-SPLINE 算法。概率生成函数嵌入到确定性算法中，以精确计算所需的性能指标，这在处理独立数据（也适用于少量产品类别）时是合理的。随机 R-SPLINE 算法使用模拟来估计所需的概率测量，从而允许不同生产配方之间以及不同需求类别之间的相关性。我们还对随机 R-SPLINE 算法进行了收敛分析。实验结果证明了所开发算法与其他现有方法相比的效率。