# Job Shop Scheduling Problem

### Sets

|  index   |  Set   |  Description | 
|:---------|:--------|:--------------|
| $m$ | $M$ | Set of nodes  |
| $j$ |  $J$   | Set of arcs  |
| $i$ | $O_j$ | Set of operations in job $j \in J$|


### Parameters

|  Parameter |  Description | 
|:---------|:--------|
| $d_{i,j}$ | Duration of operation $(i,j) \in O$ |
| $m_{i,j}$ | Machine required by operation $(i,j) \in O$|


## Discrete Time Model

Let $T$ be an upperbound on the relevant time-horizon

|  Variable | Type | Description | 
|:---------|:--------| :----- |
| $x_{i,j,t}$ | Binary | 1 if operation $(i,j) \in O$ starts at time $t \in T $|
| $z$ | continuous | Makespan  |

$$
\begin{align}
\text{minimise} \ z && & \quad (1a)\\
s.t. \ & \sum_{t \in T}t\cdot x_{i,j,t} + d_{i,j{ \leq \sum_{t \in T}t\cdot x_{i,j,t} \quad && \forall (i, j) \in A & \quad (1b) \\
& \sum_{i \in V: (i,j) \in A} f_{i, j} - \sum_{k \in V: (j, k) \in A} f_{j,k} = 0 \quad  && \forall j \in V \setminus \{s, t\} & \quad (1c) \label{m2-c2}
\end{align}
$$


## Continuous Time Model




9a3c1ba0c7b4bcca47e5873f760da0a7dc4aee78
ipynb
Jupyter Notebook
week2/.ipynb_checkpoints/Jobshop Scheduling-checkpoint.ipynb
stevedwards/gurobi_course
badb716d5dac86a77712908637cbc08722d0415d
MIT
week2/.ipynb_checkpoints/Jobshop Scheduling-checkpoint.ipynb
stevedwards/gurobi_course
badb716d5dac86a77712908637cbc08722d0415d
MIT
week2/.ipynb_checkpoints/Jobshop Scheduling-checkpoint.ipynb
stevedwards/gurobi_course
badb716d5dac86a77712908637cbc08722d0415d
MIT
Qwen/Qwen-72B
1. YES
2. YES