We consider the optimal lot size problem for multi-stage assembly systems where each facility may have many predecessors but only a single successor. Assumptions include constant continuous final product demand, instantaneous production, and an infinite planning horizon. Costs at each facility consist of a fixed charge per lot and a linear holding cost. Under the constraint that lot sizes remain time invariant, it is proven that the optimal lot size at each facility is an integer multiple of the lot size at the successor facility. This fact is used in the construction of a dynamic programming algorithm for the computation of optimal lot sizes. The algorithm exploits the concept of echelon stock [Clark, A. J., H. Scarf. 1960. Optimal policies for a multi-echelon inventory problem. Management Sci. 6 (4, July) 475-490; Clark, A. J., H. Scarf. 1962. Approximate solution to a simple multi-echelon inventory problem. Chapter 5. K. J. Arrow et al., eds. Studies in Applied Probability and Management Science. Stanford University Press, Stanford, California.].

A multi-stage assembly system is a special case of Veinott's general multi-facility system in that each facility may have any number of predecessors but at most a single successor. This paper presents two algorithms for computing optimal lot sizes in such systems with known time-varying demand. The first is a dynamic programming algorithm for which solution time increases exponentially with the number of time periods, but only linearly with the number of stages, irrespective of assembly structure. The second is a branch and bound algorithm intended for cases where the number of time periods is large but the structure is close to serial. Computational results are given and extensions considered.

The generalized linear programming algorithm allows an arbitrary mathematical programming minimization problem to be analyzed as a sequence of linear programming approximations. Under fairly general assumptions, it is demonstrated that any limit point of the sequence of optimal linear programming dual prices produced by the algorithm is optimal in a concave maximization problem that is dual to the arbitrary primal problem. This result holds even if the generalized linear programming problem does not solve the primal problem. The result is a consequence of the equivalence that exists between the operations of convexification and dualization of a primal problem. The exact mathematical nature of this equivalence is given.

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In Earth System Sciences (ESS), spatial data are increasingly used for impact research and decision-making. To support the stakeholders’ decision, the quality of the spatial data and its assurance play a major role. We present concepts and a workflow to assure the quality of ESS data. Our concepts and workflow are designed along the research data life cycle and include criteria for openness, FAIRness of data (findable, accessible, interoperable, reusable), data maturity, and data quality. Existing data maturity concepts describe (community-specific) maturity matrices, e.g., for meteorological data. These concepts assign a variety of maturity metrics to discrete levels to facilitate evaluation of the data. Moreover, the use of easy-to-understand level numbers enables quick recognition of highly mature data, and hence fosters easier reusability. Here, we propose a revised maturity matrix for ESS data including a comprehensive list of FAIR criteria. To foster the compatibility with the developed maturity matrix approach, we developed a spatial data quality matrix that relates the data maturity levels to quality metrics. The maturity and quality levels are then assigned to the phases of the data life cycle. With implementing openness criteria and matrices for data maturity and quality, we build a quality assurance (QA) workflow that comprises various activities and roles. To support researchers in applying this workflow, we implement an interactive questionnaire in the tool RDMO (research data management organizer) to collaboratively manage and monitor all QA activities. This can serve as a blueprint for use-case-specific QA for other datasets. As a proof of concept, we successfully applied our criteria for openness, data maturity, and data quality to the publicly available SPAM2010 (crop distribution) dataset series.