Late-training allocation in adolescent community Sport: Capacity, concentration and weekly rotation

Abstract

To determine the minimum number of late training sessions, limit the largest squad-level allocation and balance the remaining late appointments within a shared youth-sport calendar. Nine squads shared two courts from Tuesday to Friday, with 32 weekly 90-minute appointments starting at 16:00, 17:30, 19:00 and 20:30. Each squad required three sessions on distinct days; squads A--C were excluded from 20:30 starts. Integer optimization minimized total late sessions, maximum late sessions per eligible squad, pairwise differences in late-session counts and then summed start-time span. Ten operating conditions, all 24 individual non-late court-slot closures, equal-count quotas and a six-week rotation were evaluated. The intact calendar required three late sessions. Assigning all three to I gave a maximum of three and a Gini coefficient of 0.833; distributing one each to E, G and I gave a maximum of one and a Gini coefficient of 0.500. Four specified non-late closures required seven late sessions and a maximum of two. The vectors (1,2,0,2,2,0) and (1,2,1,1,1,1) both attained these values, but their Gini coefficients were 0.405 and 0.119. Every individual non-late closure permitted four late sessions with a maximum of one. Restricting H and I to 20:30 forced six late sessions and a maximum of three; one late session per eligible squad was infeasible. Alternating D--F and G--I over six intact weeks gave every eligible squad three late sessions. Minimizing the largest late-session count leaves the distribution among the other squads undetermined. Pairwise-difference minimization and weekly rotation balanced late appointments while retaining all requested practices. The results concern the specified scheduling restrictions and do not quantify sleep, injury or participation outcomes.