jobshop scheduling
We have
• a set of resources
• a set of jobs
• a job is a sequence of operations/activities
• sequence the activities on the resources
An example: 3 x 4
job1
Op1.1
job2
Op2.1
job3
Op3.1
•
•
•
•
•
•
Op1.2
Op2.2
Op3.2
Op1.3
Op2.3
Op3.3
Op1.4
Op2.4
Op3.4
We have 4 resources: green, yellow, red and blue
a job is a sequence of operations (precedence constraints)
each operation is executed on a resource (resource constraints)
each resource can do one operation at a time
the duration of an operation is the length of its box
we have a due date, giving time windows for operations (time constraints)
An example: 3 x 4
Op1.2
Op1.1
Op2.1
Op2.2
Op3.1
Op1.1
Op2.3
Op3.1
Op3.2
Op1.3
Op2.3
Op3.3
Op1.2
Op2.1
Op3.4
Op1.3
Op2.2
Op3.3
Op1.4
Op2.4
Op3.2
Op1.4
Op2.4
Op3.4
The problem
Assign a start time to each operation such that
(a) no two operations are in process on the same
machine at the same time and
(b) time constraints are respected
The problem is NP complete
An example: 3 x 4
Op1.1
Op2.1
Op3.1
Op1.1
Op1.2
Op2.2
Op3.2
Op1.3
Op2.3
Op3.3
Op1.4
Op2.4
Op3.4
Op2.3
Op3.1
On the “green” resource, put a direction on the arrows
A disjunctive graph
An example: 3 x 4
Op1.1
Op2.1
Op1.2
Op2.2
We do not bind operations to start times
Op3.1
Op3.2
Op1.3
Op1.4
Op2.3
Op3.3
Op2.4
Op3.4
We take a least commitment approach
Op1.1
Op2.3
Consequently we get a set of solutions!
Op3.1
On the “green” resource, put a direction on the arrows
A disjunctive graph
Apes 7x6
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What is makespan!
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;
For a long time, unsolved
Why bother?
• Minimise makespan
• what is makespan?
• Maximise start
• JIT, minimise inventory levels
• minimise idle time on resources
• maximise ROI
• ...
Op1.1
start
Op2.1
Op3.1
Op1.2
Op2.2
Op3.2
Op1.3
Op2.3
Op3.3
Op1.4
Op2.4
Op3.4
Find the smallest value for end
minimise makespan
end
How can we view this as a csp?
Each operation is a variable
domain is set of start times
there are precedence constraints between operation in a job
operations on a resource have disjunctive constraints
Op1.1
start
Op2.1
Op3.1
Op1.2
Op2.2
Op3.2
Op1.3
Op2.3
Op3.3
Op1.4
Op2.4
Op3.4
end
Complexity
What is the complexity of this problem?
•
•
•
•
Assume we have m resources and n jobs
on each resource we will have n operations
we can order these in n! ways
therefore we have O(n!m) states to explore
m
O(n! )
But we want to optimise, not satisfy
How do you optimise with CP?
A sequence of decision problems
• Is there a solution with makespan 395?
• Yip!
•
•
• Is there a solution with makespan 300?
• Let me think about that ...
• Yes
• Is there a solution with makespan 299?
• Hold on, … , hold on
• NO!
• Minimum makespan is 300.
When optimising, via a sequence of
decision problems, will all decisions be
equally difficult to answer?
What does branch and bound (BnB) do ?
Who cares about jobshop scheduling?
Manufacturing inc.
Is CP any good for this class of problem?
• Main competitor is OR technology
• Mark Fox 1980’s,
• ISIS constraint directed search(CMU)
• OPIS
• Steve Smith, Peng Si Ow (CMU)
• DAS
• Burke et al (SU)
• MicroBoss
• Norman Sadeh (CMU)
• Edge-finding
• a novel technique to infer constraints on a resources
• J Carlier & E. Pinson 1994
• CP solves open benchmarks, beats OR
• Texture based heuristics
• J-C Beck and Mark Fox 1990’s
• ILOG-Scheduler
• Claude Le Pape, Wim Nuijten, Philippe Babtiste, J-Chris Beck
• and many others
• 2000
• ILOG buy CPLEX
Variants of jsp
• openness:
• variety of resources can perform an operation
• processing time dependant on resource used
• set up costs, between jobs (transition cost)
• consumable resources
• such as gas, oil, etc
• pre-emption
• can stop and restart an operation
• resource can perform multiple operations simultaneously
• batch processing
• secondary resources
• people, tools, cranes, etc
• etc
Chris Beck (2006) “The jssp has never been spotted in the wild.”
Why might CP be technology of choice for scheduling?
• can model rich real-world problems
• addition of side constraints etc
• incorporate domain knowledge
• in the form of variable and value ordering heuristics
• powerful reasoning/inference allied to novel search techniques
We can get a solution up and running quickly
Operation (part 1)
Operation (part 2)
Resource (part 1)
Resource (part 2a)
Resource (part 2b)
Job
JSSP (part 1)
JSSP (part 2)
JSSP (part 3)
Schedule (a)
Schedule (b)
Experiment/Test
Java Schedule problems/7x6.cl
Decision Problem
Decision Problem
Wot!? No heuristics!?!!!