In this article, we present a statistical significance test for necessary conditions. This is an elaboration of necessary condition analysis (NCA), which is a data analysis approach that estimates the necessity effect size of a condition X for an outcome Y. NCA puts a ceiling on the data, representing the level of X that is necessary (but not sufficient) for a given level of Y. The empty space above the ceiling relative to the total empirical space characterizes the necessity effect size. We propose a statistical significance test that evaluates the evidence against the null hypothesis of an effect being due to chance. Such a randomness test helps protect researchers from making Type 1 errors and drawing false positive conclusions. The test is an "approximate permutation test." The test is available in NCA software for R. We provide suggestions for further statistical development of NCA.Keywords null hypothesis testing, permutation test, necessary condition analysis, statistical significance, p value Necessary condition analysis (NCA; Dul, 2016) is a tool for researchers to develop and test necessary but not sufficient conditions. A necessary condition enables the outcome when present and constrains the outcome when absent. NCA assumes that outcome Y is bound by condition X by drawing a ceiling line on top of the data in an XY scatter plot. The line defines the empty space in the upper left corner of the scatter plot.1 This empty space suggests that high values of Y are not possible with low values of X and indicates that X constrains Y. The size of the empty space relative to the total space with observations reflects the extent of the constraint that X poses on Y: The larger the empty space, the more X constrains Y. The necessity effect size (d) is the size of the empty space above the ceiling as a fraction of the total space where cases are observed or could be observed given by the minimum and maximum empirical or theoretical values of X and Y (scope 2 ). NCA's effect size d has values between 0 and 1.

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In this paper the Discrete Lotsizing and Scheduling Problem (DLSP) is considered. DLSP relates to capacitated lotsizing as well as to job scheduling problems and is concerned with determining a feasible production schedule with minimal total costs in a single-stage manufacturing process. This involves the sequencing and sizing of production lots for a number of different items over a discrete and finite planning horizon. Feasibility of production schedules is subject to production quantities being within bounds set by capacity. A problem classification for DLSP is introduced and results on computational complexity are derived for a number of single and parallel machine problems. Furthermore, efficient algorithms are discussed for solving special single and parallel machine variants of DLSP.production planning, lotsizing, sequencing, computational complexity