Scheduling and Planning

Scheduling and Planning

• Super-common application of AI search

• Already seen examples of route planning using A✶

• Drives much of business and industry today

Scheduling

• Assign start times to tasks, for example calendaring

• A task is atomic, has a name and duration

• We will consider discrete time and finite number of tasks to be scheduled

• Usually looking for optimal schedule; usually means shortest to complete all tasks

Scheduling Constraints

• Usually the task start times are constrained in some way:

  • Limit on number of tasks at a given time

  • End of task A must precede start of task B

  • Deadline by which task / tasks must be completed

  • Resources the task needs to complete

    • Capacity resource: staff, power, etc that is either enough there or isn't

    • Consumable resource: fuel, lumber, etc

Simple Case: Bin Packing

• Can run at most k tasks at a time, no other constraints

• Tasks are of varying duration: some long, some short

• This becomes a question of distributing the tasks as evenly as possible into k groups

• Special cases:

  • k = 1: Super-boring. Just run the tasks in any order end-to-start

  • k = 2: Number Partitioning. NP-hard, but most practical instances are super-fast

Simple Case: Precedence Constraints

• End-to-start precedence constraints require the end of one task to come before or at the end of another task

• Think of building a house (floor, then walls, then roof) or making a meal (boil water, make spaghetti)

• Topological sort works here:

  • Schedule tasks that can go first

  • When a task completes, schedule tasks that can now proceed

  • Continue until done

• (Can also sort end-to-start, producing a "right-justified" schedule.)

Combined Case: Resource-Constrained Project Scheduling

• If we have both parallelism constraints (think capacity resources) and precedence constraints, it is RCPS

• Tasks are marked with predecessors (come before), resource usage

• We can try topo sort, but if we don't have enough resource we can't start all the open tasks: which one?

• This can be framed as a search problem: start some subset and see how it works out

• Heuristics:

  • Short tasks first (enables more parallelism)

  • Long tasks first (since they bottleneck)

  • Hmmm

Planning

• Planning is related to scheduling, but:

  • Operators may (often will) have zero duration

  • Can perform an operation as many times as needed

  • Operations have preconditions and postconditions

Blocks World

• Standard planning environment:

  • Consider blocks A, B, ... in stacks on a table in some way

  • Want to rearrange them some different way

  • Operations:

    • Move a top block to (infinite) table to start a new stack

    • Pick up the top block of some stack and place it on top of a different stack

• There is an easy algorithm that takes at most 2n operations for n blocks…

General Purpose Planning

• Describe operators and environment in some generic way

• Have a GP planner try to find a plan

• Blocks World instances are hard for GP planners (!?)

Optimal Blocks World

• Want a shortest restacking plan

• Turns out this problem is NP-hard (!?)

• Worse yet, finding an approximately shortest plan is NP-hard (?!?)

• But can solve reasonable-sized instances by… A✶ search!

• Best planners do big state-space change to reduce search

Advanced Planning

• Logical formulation of planning problems for GP planners

• Optimal-cost planning

• Planning under uncertainty — what if:

  • Environment isn't really known for sure in the future

  • Operators might not do what they're supposed to