Mathmatically optimizing court time usage for a pickleball tournament 

Hi Michael,

You can look at this problem from different angles. "Get it done as efficiently as possible" can be minimizing the time required to complete all brackets while utilizing all courts efficiently and there is no idle time. It can also be minimizing the time required to complete all games in brackets.

We can break down the problem and formulate it mathematically as below:

$\textbf{Variables:}$

$N \text{ = number of courts available} \\$
$M \text{ = number of brackets} \\$
$G_i \text{ = number of games in bracket } i \text{, where } i = 1, 2, ..., M \\$
$T_i \text{ = time taken to complete one game in bracket } i \text{, where } i = 1, 2, ..., M \\$
$C_i \text{ = number of courts required to run bracket } i \text{, where } i = 1, 2, ..., M \\$
$S_i \text{ = start time of bracket } i \text{, where } i = 1, 2, ..., M \\$
$E_i \text{ = end time of bracket } i \text{, where } i = 1, 2, ..., M \\$
$D_i \text{ = duration of bracket } i \text{, where } i = 1, 2, ..., M \\$


$\textbf{Constraints:}$

$1. \text{ All brackets must be completed within a certain time limit (in this case, 3 hours).} \\$
$2. \text{ Each bracket must run continuously until completed once started.} \\$
$3. \text{ Brackets can run concurrently on different courts.} \\$
$4. \text{ The number of courts required for each bracket must be available.} \\$


Objective: Minimize the time taken to complete all brackets while ensuring that all courts are utilized efficiently and there is no idle time.

$\textbf{Mathematical Formulation:}$

$\text{Objective function: Minimize } \sum_{i=1}^{M} D_i \\$
$\text{Subject to:} \\$
$\quad D_i = E_i - S_i \text{ for } i = 1, 2, ..., M \\$
$\quad D_i \leq 3 \text{ hours for } i = 1, 2, ..., M \\$
$\quad S_{i+1} \geq E_i \text{ for } i = 1, 2, ..., M-1 \text{ (ensures brackets start after the previous one ends)} \\$
$\quad \sum_{i=1}^{M} C_i = N \text{ (all courts are utilized)} \\$
$\quad E_i = S_i + G_i \times T_i \text{ for } i = 1, 2, ..., M \\$


The above formulation aims to optimize the scheduling of brackets on courts while considering the time taken for each game, the number of courts available, the overall duration of the tournament and other constraints.

The optimization will ensure that all courts are utilized efficiently throughout the tournament without any idle time and maximizes the use of courts while minimizing the duration of the tournament.

Solving it will give you an optimal schedule for running your pickleball tournament efficiently. Specifically, it will provide the following:

Start and End Times for Each Bracket, Number of Courts Needed for Each Bracket, Overall Duration of the Tournament, and Minimized Time to Complete All Brackets.

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You can also model the problem considering the time taken for each bracket, the number of courts available, and the overall duration of the tournament. See below:

$M: \text{Set of brackets} \\$
$G_i: \text{Number of games in bracket } i \\$
$C: \text{Set of courts} \\$
$t_{ic}: \text{Time it takes to complete bracket } i \text{ on court } c \\$
$x_{ic}: \text{Binary variable indicating whether bracket } i \text{ is scheduled on court } c$


$\text{Minimize: } \sum_{i \in M} \sum_{c \in C} t_{ic} \cdot x_{ic}$

Constraints:

$\text{1. Each game in a bracket takes 20 minutes, and each bracket must complete within 3 hours:} \\$
$\sum_{c \in C} G_i \cdot x_{ic} \leq \frac{3 \times 60}{20} \quad \forall i \in M \\$
$\text{2. Each court can only host one bracket at a time:} \\$
$\sum_{i \in M} x_{ic} \leq 1 \quad \forall c \in C \\$
$\text{3. Every bracket must be scheduled:} \\$
$\sum_{c \in C} x_{ic} = 1 \quad \forall i \in M \\$
$\text{4. Non-negativity constraints:} \\$
$x_{ic} \geq 0 \quad \forall i \in M, \forall c \in C$


Both formulations are correct and essentially represent the same problem but with different decision variables and constraints. The decision of which formulation to use depends on how you specifically want to implement this and what exactly you had in mind.