混合泳接力最优分配

Making a plan

I’m organizing swimmers for a medley relay to cut down total time. Their times in backstroke, breaststroke, butterfly, and freestyle show their pace.

Tackling combinatorial options

I’m thinking through all swimmer permutations for strokes to find the optimal total time. Each stroke is unique to each swimmer, ensuring a fair and balanced distribution.

Mapping out permutations

I’m totaling times for different swimmer orderings. Each setup is carefully checked to confirm its accuracy. The sum for this particular setup is 276.5.

Evaluating permutations

Checking swimmer combinations for stroke sequences reveals total times of 261.6, 273.8, 273.4, 261, 275.5, and more, pinpointing the best option.

Mapping out options

Summing swimmer times reveals a total of 287.4. Pulled together, varied permutations result in totals of 272.5, 273.7, 269.3, 260.9, and more.

Evaluating swimmer setups

Adding up the eyelash dye times for combinations like (C,B,D,A) totals 253, while (C,B,A,D) yields 257.4. Progressing through combinations to find the optimal time.

Assessing swimmer orders

I’m noting swim combinations like (D,A,C,B) totaling 291.4, and (D,B,A,C) at 265.9. Progress is evident in refining the optimal swimmer arrangement for the medley relay.

Evaluating outcomes

I’ve pinpointed the fastest medley relay swimmer combination, with a total time of 252.4 seconds for permutation (C,D,B,A). This option outperforms others in efficiency.

Confirming the best lineup

I’ve verified the best medley relay swimmer combination, totaling 252.4 seconds. Double-checking calculations showed no discrepancies. The second-best lineup clocks in at 253 seconds.

Thought about mixed relay selection，持续 22 秒

解题思路大致如下：

1. 明确接力顺序及分配原则

   • 4×100 米混合泳接力的出场顺序是：仰泳 → 蛙泳 → 蝶泳 → 自由泳。
   • 甲、乙、丙、丁四名运动员恰好要各负责其中一种泳姿，且每人只能游一种姿势。

2. 从表格中读取各运动员在四种泳姿下的成绩
   设四名运动员分别为 A(甲)、B(乙)、C(丙)、D(丁)。表格可简化为：

   运动员	仰泳(Bk)	蛙泳(Br)	蝶泳(Fly)	自由泳(Fr)
   A(甲)	75.5	86.8	66.6	58.4
   B(乙)	65.8	66.2	57.0	52.8
   C(丙)	67.6	84.3	77.8	59.1
   D(丁)	74.0	69.4	60.8	57.0

3. 列举（或程序穷举）所有可能的分配方式

   • 因为有 4 位运动员和 4 个泳姿，需要考虑 4! = 24 种不同的分配/排列。
   • 对每种分配方式，分别取对应的用时相加，得到该分配方式下的总用时。

4. 计算并比较每种分配方式的总时间，找出最优解
   在 24 种分配方案中，找到总用时最少的一组。经过计算（此处只展示最优方案的由来），最优分配是：

   • 仰泳：C(丙) → 67.6 秒
   • 蛙泳：D(丁) → 69.4 秒
   • 蝶泳：B(乙) → 57.0 秒
   • 自由泳：A(甲) → 58.4 秒

   四段合计用时：

   $$67.6 + 69.4 + 57.0 + 58.4 \;=\; 252.4 \;\text{(秒)}$$

   这是所有排列中总用时最短的组合。

答案

• 最佳安排：
  • 仰泳：丙
  • 蛙泳：丁
  • 蝶泳：乙
  • 自由泳：甲
• 合计用时：252.4 秒

这样分配可以使接力队的 4×100 米混合泳总成绩达到最优。