JobShopScheduling/doc.md

scheduling problem defined by:

1. m specialized machines
2. tasks \tau of the form (e, i) with t \in \mathbb{N} the execution time and i \in \{1,2,\dots,m\} the machine the task has to run on
3. n jobs T_k with \forall T_k: linear order of tasks, with k \in \{1,2,\dots,n\}
4. Additionally, a multiset \Omega of arbitrary but fixed size that contains wait states \omega := (1, i) with i \in \{1,2,\dots,m\} the blocked machine.

The goal is to find the fastest feasible schedule \sigma_{min}.

evaluative function

• minimize the execution time of \sigma
• upper bound: largest processing time first
• lower bound: max sum of execution times on one machine

solutions

• list of tuples (t, \tau) with t \in \mathbb{N} the scheduled begin of \tau

operations

• \operatorname{ins}(\omega, t): block a machine at time t for w time steps.
• \operatorname{xchg}(\tau_1,\tau_2): exchange the position of two tasks.

Both operations require that the start times are recomputed.

neighbourhood of solution

• \operatorname{neighbours}(\sigma) = \{x \in \Sigma | \delta(\sigma, x) \leq n\} with \Sigma the set of all feasible schedules.
• \delta: \delta ( \sigma )=0, \delta ( \operatorname{op}(x)) = \delta (x) + 1 (ass. ins has the same penalty xchg has), x either op(y) or \sigma

constraints

• only schedule new \tau if another \tau is finished
• only schedule \tau \in T_k that has no unscheduled predecessor in T_k
• only one task on a machine any given time

implementation in Python

• translate problem into list of jobs, jobs into lists of tasks, ie problem = [$T_0, T_1,\dots,T_{k-1}$], T_i = [$\tau_1,\tau_2,\dots$]
• address tasks based on their indices, ie [0][1] is the second task of the first job.
• compute only one possible next solution, rate, drop/accept. \delta is computed iteratively during generation