This paper presents two novel formulations for scheduling problems, namely order-position hybrid formulation (OPH) and order-disjunctive hybrid formulation (ODH), which extend and combine parts of existing formulation strategies. The first strategy (OPH) is based on sequence position and linear ordering formulations, adding relationships between constraints that allow relaxing some decision variables. The second approach (ODH) is based on linear ordering and disjunctive formulations. In this work, we prove ODH to be the most efficient formulation known so far. The experiments have been carried out with a large set of problems, which consider single machines and identical parallel machines. Computational results show that OPH is better than the rest of the existing formulations for the case of weighted completion objectives, while ODH turns out to be the best approach for most scenarios studied.

This paper presents a strategy based on binary labelling of nodes for the creation of anti-loop formulations from existing strategies. This strategy prevents by default the formation of odd cycles, therefore it can have important role in iterative procedures based on generating subtour elimination constraints. It can also be used to modify the classic strategies used in problems associated to graphs. In this paper we focus on this last application. The behavior of this strategy is analyzed with two problems associated with graphs, the Asymmetric Traveling Salesman Problem (ATSP) and the Steiner Problem, where two configurations that modify the Miller-Tucking-Zemlig proposal to avoid cycles are compared. The experimental analysis shows that this strategy keep a good convergence, highlighting its use for the Steiner problem.

Introduction: Yoga is a training methodology that may facilitate the physical and psychological restoration of football players. Objectives: To evaluate the effects of Yoga on football-related abilities, based on the hypothesis that Yoga training would not negatively affect speed, agility, or change of direction (COD). Yoga is a practical life philosophy aimed at promoting self-knowledge through systematic training of body and mind. It may positively influence players’ change-of-direction deficit due to its physical and psychological effects on recovery processes. Although Yoga offers a holistic approach to recovery, concerns remain regarding a potential reduction in performance associated with static stretching. Methodology: A convenience sample of 21 professional football players from the Portuguese First League participated (age: 19.0 ± 1.89 years; height: 182.5 ± 6.02 cm; body mass: 75.55 ± 5.69 kg). Participants were randomly allocated to an experimental group (EG; n = 11) or a control group (CG; n = 10). The EG completed a 12-week Yoga training programme (two additional 60-minute sessions per week), including prāṇāyāma, āsana, yoga nidrā and samyama, while the CG followed standard football training. Sprint speed (5, 10 and 20 m) and agility (zigzag test) were assessed pre- and post-intervention using photocell timing gates. Mixed-factor ANOVA and effect sizes (partial eta squared) were calculated, with significance set at p < 0.05. Results: Both groups showed significant improvements from pre- to post-test across all variables. No significant group × time interactions were found for 5 m, 10 m, 20 m sprint, zigzag agility or COD deficit, indicating that Yoga did not impair performance. Conclusions: Yoga represents a viable alternative to enhance mobility and support physical and psychological recovery without negatively affecting sprint speed, agility or COD in football players.