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We consider optimization problems expressed in Decision Guidance Query Language that may involve linear arithmetic constraints, as well as finite domain and binary variables, and focus on a class of Multi-Stage Production problems in which only a part of the problem is dynamic, i.e., the demand for the output product in a manufacturing process, whereas the rest of the problem is static, i.e., the connectivity graph of the assembly processes and the cost functions of machines. We propose the online-decomposition algorithm (ODA) based on offline preprocessing that optimizes each static problem component for discretized values of shared constraint variables, and approximate the optimal aggregated utility functions. ODA uses the pre-processed approximated aggregated cost functions to decompose the original problem into smaller problems, and utilizes search heuristics for the combinatorial part of the problem based on the pre-processed look-up tables. We also conduct an initial experimental evaluation which shows that ODA, as compared with MILP, provides an order of magnitude improvement in terms of both computational time and the quality of found solutions for a class of problems for which preprocessing is possible.