Advances in solving large scheduling problems with CP Optimizer

Feb. 20, 2019
Feb. 20, 2019
ROADEF-2019
ROADEF-2019
© 2019 IBM Corporation
© 2019 IBM Corporation
Recent advances on
large scheduling problems
in CP Optimizer
Philippe Laborie, IBM
laborie@fr.ibm.com

2 / 17 ROADEF 2019 © 2019 IBM Corporation
In this presentation
1. Recap about IBM CP Optimizer
By the way: if you have CPLEX, you also have
CP Optimizer!
2. Performance improvements on scheduling problems
in coming release V12.9

CP Optimizer for scheduling in two sentences
A mathematical modeling language for combinatorial
optimization problems that extends ILP (and classical CP)
with some algebra on intervals and functions allowing
compact and maintainable formulations for complex
scheduling problems
A continuously improving automatic search algorithm
that is complete, anytime, efficient (e.g. competitive with
problem-specific algorithms on classical problems) and
scalable
Recent review of CP Optimizer (modeling concepts,
applications, examples, tools, performance,…) :
IBM ILOG CP Optimizer for scheduling. Constraints (2018)
23:210–250. http://ibm.biz/Constraints2018

Typical scheduling problems for CP Optimizer

Illustration on an academical scheduling problem
Resource-Constrained Project Scheduling (RCPSP)
Notorious NP-Hard problem in combinatorial optimization
(>5000 references on Google Scholar)
N tasks with precedence constraints
M resources of limited capacity
Minimize project makespan
1
1
2
2
4
4
3
3
5
5
6
6
0 7
R1
R2
1
2
3
4
5
6
p1
=2
(1,2)
C1
=2
C2
=3
time

Formulation of the RCPSP in ILP
Example of the Start/End Event-based formulation (SEE)
O(N3
)
* A. Tesch. Compact MIP Models for the Resource-Constrained Project Scheduling
Problem. Master's Thesis. Technische Universität Berlin. 2015.

Formulation of the RCPSP in CP Optimizer
Use of interval variables for tasks and cumul functions
for resource usage
O(N)
xi
xj
Qij
Di
pulse(xi
,Qij
)
compact

Formulation of the RCPSP in CP Optimizer
Use of interval variables for tasks and cumul functions
for resource usage
from docplex.cp.model import *
model = CpoModel()
# Decision variables: tasks x
x = [interval_var(size=D[i]) for i in N]
# Objective: minimize project makespan
model.add(minimize(max(end_of(x[i]) for i in N)))
# Constraints: resource capacities
for j in M: model.add(sum(pulse(x[i],q) for [i,q] in R[j]) <= C[j])
# Constraints: precedence between tasks
for [i,j] in P: model.add(end_before_start(x[i],x[j]))
# Solve the model
sol = model.solve(TimeLimit=300)
compact
xi
xj
Qij
Di
pulse(xi
,Qij
)

Classical RCPSP benchmarks (5023 instances)
CP Optimizer 12.9 results with automatic search using a
time limit of 5mn (code of previous slide)
Maximal time to first feasible solution: 0.08s anytime
complete
efficient

Classical RCPSP benchmarks
CP Optimizer 12.9 results with automatic search
completeefficient

Classical RCPSP benchmarks
CP Optimizer 12.9 results with automatic search using a
time limit of 5mn (code of previous slide)
These problems are tiny compared to industrial problems !

Automatic Search
Main principle: cooperation between several approaches
100
101
102
103
104
105
106
CP Optimizer Automatic Search
Failure-Directed Search
Large-Neighborhood Search
Iterative Diving
Problem size
(number of
interval
variables)
Existing RCPSP instances

Iterative diving
By the way: “CP” in “CP Optimizer” means Constraint
Programming 
Idea of iterative diving: perform aggressive dives (no
backtrack) in the search tree explored by CP
For instance on RCPSP it boils down to some very
classical ideas (list scheduling, decoding schemes, …)
The challenge was to generalize these problem-specific
ideas to the general modeling concepts of CP Optimizer

CP Optimizer V12.9
New benchmark with RCPSP from 500 to 500.000 tasks
Largest problem: 500.000 tasks, 79 resources,
4.740.783 precedences, 4.433.550 resource requirements
Time to first feasible solution (V12.8 v.s. V12.9)
0E+00 5E+04 1E+05 2E+05 2E+05 3E+05 3E+05 4E+05 4E+05 5E+05 5E+05
0
500
1000
1500
2000
2500
3000
First solution time for large RCPSPs (automatic search with 4 workers)
V12.8 V12.9
Instance size (Number of tasks)
Firstsolutiontime(s)
scalable

Example of cumul function value
ResourceloadResourceloadResourceload
Resource load for one of the resources of a large instance
(500.000 tasks)

CP Optimizer automatic search
Objective
landscapes
Failure-directed
search
Iterative
diving
With 4 workers, average speed-up of 42% between 12.8
and coming version 12.9

Conclusion
A mathematical modeling language for combinatorial
optimization problems that extends ILP (and classical CP)
with some algebra on intervals and functions allowing
compact and maintainable formulations for complex
scheduling problems
A continuously improving automatic search algorithm
that is complete, anytime, efficient (e.g. competitive with
problem-specific algorithms on classical problems) and
scalable
Recent review of CP Optimizer (modeling concepts,
applications, examples, tools, performance,…) :
IBM ILOG CP Optimizer for scheduling. Constraints (2018)
23:210–250. http://ibm.biz/Constraints2018

Advances in solving large scheduling problems with CP Optimizer

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