We consider the online (over time) scheduling on a single unbounded parallel-batch machine with job processing time compatibilities to minimize makespan. In the problem, a constant \(\alpha >0\) is given in advance. Each job \(J_{j}\) has a normal processing time \(p_j\). Two jobs \(J_i\) and \(J_j\) are compatible if \(\max \{p_i, p_j\} \le (1+\alpha )\cdot \min \{p_i, p_j\}\). In the problem, mutually compatible jobs can form a batch being processed on the machine. The processing time of a batch is equal to the maximum normal processing time of the jobs in this batch. For this problem, we provide an optimal online algorithm with a competitive ratio of \(1+\beta _\alpha \), where \(\beta _\alpha \) is the positive root of the equation \((1+\alpha )x^{2}+\alpha x=1+\alpha \). 相似文献
Industrialized areas represent a special group of urbanized areas, but sufficient attention has not been given to them concerning plant species richness and its importance for biodiversity conservation. These sites are considered the main drivers of biological invasions and biotic homogenization on a global scale, but the question of how and to what degree they contribute to biodiversity has yet to be raised. Data for 62 river ports (16 German, 26 Czech, 7 Hungarian, 4 Slovak, and 9 Austrian) on two important Central European waterways (the Elbe-Vltava and Danube waterways) were gathered for 40 years. In total, 1 240 plant species were found. Of these, 371 were classified as threatened and protected, nearly one third of the total number of species found in all of the studied Elbe, Vltava, and Danube ports. Significant differences in the proportion of threatened species in ports and cities were not found. More threatened species were found in the Danube ports than in the other ports, but their numbers fluctuated depending on the size of the port and the region (country). The results showed that the ports must be regarded as special types of species-rich industrial areas; thus, they should not be considered to be only sources of spreading invasive alien species. When planning port development, uniformity and vegetation management that is too intensive should be avoided. Extensively managed open areas and habitat diversity should be maintained.