Managing Swimmer Workload & Fatigue in Multi-Event Meets

1. Fatigue Factors in Swim Meets Multi-event meets present scheduling challenges. When a swimmer is entered in consecutive relays without adequate rest, their split times decay due to lactic acid accumulation. In Masters meets, workload control is essential to prevent injury.

During high-intensity exercise like swimming, the body relies on anaerobic glycolysis for energy, which produces lactic acid as a byproduct. This leads to a build-up of hydrogen ions in the muscles, causing acidosis and reducing the muscles' ability to contract forcefully. In addition, repeated sprints deplete the muscles' stores of glycogen and creatine phosphate.

For older athletes in masters swimming, the recovery time between events is significantly longer than for younger swimmers. A 55-year-old athlete may need 30 to 45 minutes of active recovery (warm-down) to return to baseline performance, whereas an 18-year-old might recover in 15 minutes. If a coach schedules an older swimmer for back-to-back relays with only a 10-minute gap, their performance in the second relay will suffer dramatically, and they face a heightened risk of muscle strains or shoulder injuries.

2. Setting Workload Limits Our planner models workload boundaries by assigning a maximum relay count constraint ($M_i$) to each swimmer $i$. The assignment variables must satisfy:

$$\sum_{e \in E} \sum_{j \in J} x_{i,j,e} \leq M_i \quad \forall i \in I$$

where $E$ is the set of all relay events in the meet.

This constraint is crucial for maintaining team health and ensuring that no single athlete is over-extended. In a typical two-day championship meet, there might be 6 or 8 different relay events (including gender-specific and mixed relays of various distances). Without a strict workload limit, the optimizer might assign the team's star athlete to all 8 relays, as they are likely the fastest option for many slots.

By enforcing a workload cap (e.g., $M_i = 3$ or $M_i = 4$), the coach protects the athlete and forces the optimizer to find alternative configurations. This makes the optimization problem much more interesting, as the solver must decide which events are the highest priority for the star athlete, and where the secondary swimmers can be placed to maximize the team's overall score.

3. Decay Modeling A swimmer's expected split time $t_{i,j,e}$ increases if the swimmer was assigned to another event $e'$ scheduled shortly before $e$. We represent this fatigue decay factor as:

$$t_{i,j,e} = t_{i,j}^0 \times (1 + \delta(e - e'))$$

where $t_{i,j}^0$ is the baseline seed time and $\delta$ is the fatigue penalty coefficient.

The penalty coefficient $\delta(d)$ is a function of the distance $d$ between the two events. If $d = 1$ (meaning the events are consecutive), the penalty is highest (e.g., a 2% to 5% increase in swim time). If $d \geq 4$, the penalty decays to zero, assuming the swimmer has had sufficient time to warm down, rest, and rehydrate.

```typescript function calculateFatiguePenalty(baseTime: number, eventGap: number): number { if (eventGap <= 0) return baseTime; // Exponential decay model for fatigue penalty const maxPenaltyRate = 0.05; // 5% maximum penalty for back-to-back events const decayRate = 0.5; // Penalty halves with each event gap const penalty = maxPenaltyRate * Math.exp(-decayRate * (eventGap - 1)); return baseTime * (1 + penalty); } ```

This mathematical model allows the optimizer to make realistic predictions. For example, if a swimmer's baseline 50 Free time is 25.0 seconds, a back-to-back entry would result in a predicted time of 26.25 seconds. The optimizer might then decide that a rested swimmer with a baseline of 26.0 seconds is actually the faster option for that second relay.

4. Roster Rotations and Team Depth By enforcing strict workload boundaries and accounting for fatigue decay, the solver forces the rotation of bench swimmers into secondary (B and C) teams. This improves team morale by ensuring all roster members participate while safeguarding peak athletes for primary record-breaking events.

In large swim clubs, team culture is just as important as winning. Swimmers who train hard all season want to compete in relays, not just watch from the stands. By using an optimization model that manages workloads, coaches can demonstrate to their team that relay selections are fair, objective, and designed to maximize both participation and performance.

Furthermore, this approach helps coaches build long-term team depth. By giving secondary swimmers the opportunity to swim in competitive relays, they gain valuable racing experience and improve their skills. Over time, these swimmers develop into key contributors, making the team stronger and more resilient to injuries or absences.