Due to unforeseen events (e.g. bad weather conditions), association football league schedules are not necessarily played as they were announced in the beginning of the season. This paper analyses the impact of uncertainty on the quality of football league schedules by examining fifteen seasons of ten major European football leagues. We describe several quality measures, related to breaks, the fairness of the ranking, and cancelled matches. The empirical study reveals that matches that were rescheduled to another date have a profound impact on the quality of the resulting schedule, indicating that football schedules in Europe deal poorly with uncertainty. Moreover, we present several proactive and reactive approaches in order to mitigate this problem. The former determine where to insert so-called catch-up rounds as buffers in the schedule, while the latter reschedule matches to these catch-up rounds when uncertain events occur. We evaluate combinations of proactive and reactive approaches, and provide recommendations to practitioners (e.g. four catch-up rounds usually suffice, and immediate irrevocable rescheduling is not beneficial).

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Fuzzy clustering is a widely used approach for data classification by using the fuzzy set theory. The probability measure and the possibility measure are two popular measures which have been used in the fuzzy [Formula: see text]-means algorithm (FCM) and the possibilistic clustering algorithms (PCAs), respectively. However, the numerical experiments revealed that FCM and its derivatives lack the intuitive concept of degree of belongingness, and PCAs suffer from the “coincident problem” and cannot provide very stable results for some data sets. In this study, we propose a new clustering algorithm, called the credibilistic clustering algorithm (CCA), based on the credibility measure. The credibility measure provides some unique properties which can solve the “coincident problem” and noise issue compared with the probability measure and possibility measure. Based on some randomly generated data sets, experimental results compared with FCM and PCA show that CCA can deal with the “coincident problem” with good clustering results, and it is more robust to noise than PCA.

Variance is of great significance in measuring the degree of deviation, which has gained extensive usage in many fields in practical scenarios. The definition of the variance on the basis of the credibility measure was first put forward in 2002. Following this idea, the calculation of the accurate value of the variance for some special fuzzy variables, like the symmetric and asymmetric triangular fuzzy numbers and the Gaussian fuzzy numbers, is presented in this paper, which turns out to be far more complicated. Thus, in order to better implement variance in real-life projects like risk control and quality management, we suggest a rational upper bound of the variance based on an inequality, together with its calculation formula, which can largely simplify the calculation process within a reasonable range. Meanwhile, some discussions between the variance and its rational upper bound are presented to show the rationality of the latter. Furthermore, two inequalities regarding the rational upper bound of variance and standard deviation of the sum of two fuzzy variables and their individual variances and standard deviations are proved. Subsequently, some numerical examples are illustrated to show the effectiveness and the feasibility of the proposed inequalities.

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