Modeling and Solving Scheduling Problems with CP Optimizer

© 2014 IBM Corporation
May 28, 2014
IBM Decision Optimization Virtual User Group Meeting
Modeling and Solving Scheduling Problems
with CP Optimizer
Philippe Laborie
CPLEX Optimization Studio Team

© 2014 IBM Corporation2
Agenda
 IBM ILOG CP Optimizer for Scheduling
 Scheduling concepts in CP Optimizer
 Tips and Tools
 Q&A

IBM ILOG CP Optimizer for Scheduling
 Scheduling consist of assigning starting and completion times
to a set of activities while satisfying different types of constraints
(resource availability, precedence relationships, … ) and
optimizing some criteria (minimizing tardiness, …)
 Time is considered as a continuous dimension: domain of possible
start/completion times for an activity is potentially very large
 Beside start and completion times of activities, other types of
decision variables are often involved in real industrial scheduling
problems (resource allocation, optional activities …)
start end
Timeactivity

 Modeling and solving scheduling problems
Model & run
paradigm
User-implemented
resolution
dispatching rules,
heuristics,
meta-heuristics
MIP
(CPLEX)

 Modeling and solving scheduling problems
– Dedicated modeling concepts for scheduling
• Consistent with Optimization paradigm:
new decision variables, expressions and constraints
• Available in OPL and APIs (C++, Java, .NET)
– Automatic search
• Complete
• Combines tree search, large neighborhood search, meta-
heuristics, relaxations, learning, ...
Model & run
paradigm
User-implemented
resolution
dispatching rules,
heuristics,
meta-heuristics
MIP
(CPLEX)
CP Optimizer
for Scheduling

Agenda
 Tips and Tools
 Q&A

Concept: interval variable
 What for?
–modeling an interval of time during which a particular property
holds (an activity executes, a resource is idle, a tank must be
non-empty, …)
 Example:
dvar interval x in 0..1000 size in 10..20
0 1000
Time
[10,20]
Interval variable

Concept: interval variable
dvar interval x in 0..1000 size in 10..20
 Properties:
–The value of an interval variable is an integer interval
[start,end)
–Domain of possible values: [0,10), [1,11), [2,12),...
[990,1000), [0,11),[1,12),...
–Domain of interval variables is represented compactly in CP
Optimizer (a few bounds: smin, smax, emin, emax, szmin,
szmax)
0 1000
Time
[10,20]
Interval variable

Concepts: optional interval variable
 Interval variables can be defined as being optional that is, it is
part of the decisions of the problem to decide whether the
interval will be present or absent in the solution
 What for?
–Modeling optional activities, alternative execution modes for
activities, and … most of the discrete decisions in a schedule
 Example:
dvar interval x optional in 0..1000
size in 10..20
 Properties:
–An optional interval variable has an additional possible value in
its domain (absence value)
–Optionality is a powerful property that you must learn to
leverage in your models (more on this later ...)

Concept: precedence constraint
 What for ?
–Modeling temporal constraints between interval variables
–
 Example:
a b endBeforeStart(a,b)
startBeforeStart(a,b)
startBeforeEnd(a,b)
endBeforeEnd(a,b)
endAtStart(a,b)
startAtStart(a,b)
startAtEnd(a,b)
endAtEnd(a,b)

 What for ?
–Modeling temporal constraints between interval variables
–Modeling constant or variable minimal delays
 Example:
a b endBeforeStart(a,b,z)
startBeforeStart(a,b,z)
startBeforeEnd(a,b,z)
endBeforeEnd(a,b,z)
endAtStart(a,b,z)
startAtStart(a,b,z)
startAtEnd(a,b,z)
endAtEnd(a,b,z)
z

 Properties
–Semantic of the constraints handles optionality (as for all
constraints in CP Optimizer). Example of endBeforeStart:
present(a) AND present(b)  end(a)+z  start(b)
–All precedence constraints are aggregated in a temporal
network and handled by dedicated graph algorithms (fast
global propagation, negative cycle detection, ...)

Concept: presence constraint
 What for:
–Expressing dependency constraints between execution of
optional activities or between allocated resources …
 Examples:
a b presenceOf(a) => presenceOf(b)
presenceOf(a) == presenceOf(b)
presenceOf(a) => !presenceOf(b)
!presenceOf(a) => presenceOf(b)

Concept: presence constraint
 Properties:
–All binary constraints between presence status are aggregated in
an implication graph
–CP Optimizer maintains hypothetical bounds on interval variables
(e.g. start min would the interval be present)
–CP Optimizer exploits the implication graph to perform
conditional reasoning between related interval variables

Concept: expressions on interval variables
 What for?
– Evaluating a characteristic of an interval variable (start, end,
size) for use in the objective function or in a constraint
 Example:
– endOf(a, ABSVAL) takes value ABSVAL if interval a is absent
otherwise it takes value e if a is present and a=[s,e)
–Typical makespan expression: max(i in 1..n) endOf(a[i])

Concept: expressions on interval variables
 What for?
–Evaluating a piecewise linear function at a particular end-point
of an interval variable (earliness-tardiness cost, temporal
preference)
 Example:
– endEval(f,a, ABSVAL) takes value ABSVAL if a is absent
otherwise it takes value f(e) if a is present and a=[s,e)
 Property:
–f can be any piecewise linear function (non-continuous, non-
convex, ...)
f
a

Concept: cumul function
 What for?
–Modeling and constraining cumulative quantities over time
(cumulative use of a discrete resource, level of an inventory with
producing and consuming tasks)
–Restricting the number of intervals that overlap a given date t
–Forcing a minimal number of intervals to overlap a given date t
–Complex synchronization constraints between interval variables:
e.g. between activities producing/consuming in a tank and the
set of time-intervals during which the tank is non-empty

 Cumul functions are built from atomic functions




 A cumul function is the sum of atomic functions or their opposite
a
pulse(a,h)
h
a
stepAtStart(a,h)
h
a
stepAtEnd(a,h)
h

 Examples:
–cumulFunction f = pulse(a,2)+pulse(b,1)+pulse(c,3)
–
–
–
–
–
–
–
–Constraint f<=C (global capacity limit)
–Constraint alwaysIn(f,t0,t1,Cmin,Cmax) (capacity profile)
–Constraint alwaysIn(f,x,Cmin,Cmax) (condition holds over an
interval variable x)
a
b c
f

 Properties:
–Complexity of cumul functions is independent of the time
scale
–CP Optimizer is able to reason globally over cumul functions

 Properties:
–Compare:
• Time-indexed MIP model:
• s[i][t]{0,1}: s[i][t]=1 iff a[i] starts at date t
•
•
•
•
•
•
• CP Optimizer model:
a[i]
t
t
sum(i in …) pulse(a[i],Q[i])  C
t, sum(i in …, t' in …) Q[i]*s[i][t'] C
...

 Example: Resource-Constrained Project Scheduling Problem
(RCPSP)
 Minimization of project makespan
a1
a3
a2
a6
a5
a4
a7
a8
R1
R2

 CP Optimizer model for RCPSP:
 Comparison of this CP Optimizer model vs a time-indexed MIP
model on a set of 300 classical small RCPSP instances (30-120
tasks) + 40 larger ones (900 tasks), time-limit: 2mn, 4 threads

 Comparison of CP Optimizer and MIP performance on RCPSP

Concept: sequence variable
 What for?
–modeling sequences of events
–constraining the transitions between consecutive events (setup
times/costs,...)
 Example:
dvar sequence s in all(i in Tasks) a[i]
noOverlap(s)
 Properties:
– A value for the sequence variable is a total order on the present
interval variables. E.g. a4 a1 a3 a2→ → →
–A constraint noOverlap states that intervals of the sequence do
not overlap and follows the total order of the sequence
a4 a1 a3 a2 Time

 Properties:
–Complexity is independent of the time scale
–CP Optimizer is able to reason globally over a sequence variable
–Avoid quadratic models over each pair (i,j) of intervals in the
sequence
 Compare:
– Quadratic disjunctive MIP formulation with big-Ms:
b[i][j]{0,1}: b[i][j]=1 iff a[i] before a[j]
end[i] <= start[j] + M*(1-b[i][j])
end[j] <= start[i] + M*b[i][j]
– CP Optimizer model:
a[i]
Time
noOverlap(all(i in …) a[i])
a[j]
…

 Example: Job-shop Scheduling Problem
 Minimization of makespan
op11 op12 op13
M1
M2
M3
op21 op22 op23
op31 op32 op33
op41 op42 op43

 CP Optimizer model for Job-shop:
 Comparison of this CP Optimizer model vs a disjunctive MIP
formulation on a set of 140 classical Job-shop instances (50-2000
tasks), time-limit: 2mn, 4 threads

 Comparison of CP Optimizer and MIP performance on Job-Shop

Concept: state function
 What for?
–Modeling evolution of a state variable over time
–Modeling incompatibility relations between activities
overlapping a given date t
–Synchronizing start/end value of compatible tasks (batching)
 Example
 Properties
–Complexity is independent of the time scale
–CP Optimizer is able to reason globally over a state function
–Avoid quadratic models over each pair (i,j) of incompatible
intervals on the state function
Time400˚F 500˚F 500˚F 600˚F

Concept: span constraint
 What for?
–Modeling task sub-task decomposition→
–Modeling immobilization time of a resource
 Example
span(a,[b1,...,bn])
 Properties
–The constraint of course handles optional interval variables
a
b3
b4
b1
b2

Concept: alternative constraint
 What for?
–Modeling alternative resource/modes/recipes
–In general modeling a discrete selection in the schedule
 Example
alternative(a,[b1,...,bn])
 Properties
–Conditional reasoning allows a strong pruning on the
alternative master interval variable a
–Master interval variable a can of course be optional
a
b3
b1
b2
Absent interval variables
Selected interval variableALT

Concept: span/alternative constraint
 What for?
–Modeling Work Breakdown Structure
 Example
Task
Recipe 2Recipe 1 Recipe 3
ALT
op1 op2
op3
ALT

Agenda
 Tips and Tools
 Q&A

Tips and Tools
 A measure of complexity of a model:
–Number of variables
–Number of groups of variables with different semantics
–Number of variable types (integer variables / interval variables)
 Take advantage of the expressiveness of the CP Optimizer
modeling language to compact the model by decreasing the
above indicators
 In particular if in a scheduling model, you are creating 1 decision
variable for each time point t you are usually in bad shape
➔
Exploit the scheduling concepts to reason on time-lines (sequence
variables, cumul functions, state functions) and avoid an explicit
representation of time

Tips and Tools
 A measure of complexity of a model is the number of
constraints
 If the number of constraints grows more than linearly with the
number of variables or the size of variables domains, the model
will probably not scale well
 Furthermore, if a set of many small constraints can be
reformulated more compactly, this often leads to stronger
inference in the engine
 Example: when you need to model activities (and more generally
intervals of time) that -under some conditions- cannot overlap,
think of using sequence variables, noOverlap constraints and/or
state functions

Tips and Tools
 Be careful when using composite constraints
 In scheduling models, exploit optionality of interval variables
 Examples :
– presenceOf(a)*endOf(a)+(1-presenceOf(a))*K USE: endOf(a,K)
– presenceOf(a) => (10<=startOf(a)) USE: 10<=startOf(a,10)
 Remarks:
–Exception: Binary logical constraints on presence status (like
presenceOf(a) => presenceOf(b)) are handled in a special
(and efficient) way in the engine
–Except for constraint “presenceOf”, all constraints on
scheduling constructs (interval and sequence variables, cumul
and state functions) cannot be used in composite constraints

Tips and Tools
 Surrogate constraints may help to reduce the search space
 A surrogate constraint is a constraint that does not change the
set of solutions (but may be useful for the search by allowing a
better pruning of the search space)
 Some examples :
–Redundant cumul function for alternative resources

Tips and Tools
 Surrogate constraints may help to reduce the search space
 A surrogate constraint is a constraint that does not change the
set of solutions (but may be useful for the search by allowing a
better pruning of the search space)
 Some examples :
–Redundant cumul function for alternative resources
–Redundant precedence constraints
a
a2
a1
ALT
b
b2
b1
ALT
z11
z22
z12 z21
min(z11,z12,z21,z22)

Tips and Tools
 Some classical patterns in scheduling models
–Chain of optional interval variables
–Optional intervals in paths of precedence constraints
–Multilevel alternatives
–Temporal alternatives
 See bonus material at the end of the slide deck

Tips and Tools
 Questions:
–Why is my model infeasible ?
–Search does not find any feasible solution, what should I do ?
–Search does not converge fast enough, what should I do ?
–Does my model improvement really help?

Tips and Tools
 Why is my model infeasible ?
–Conflict refiner Fix/relax model→

Tips and Tools
 The engine log is an essential tool for understanding how the
engine behaves on a given model
! -----------------------------------------------------------------------
! Minimization problem - 110 variables, 100 constraints
! LogPeriod = 100,000
! Workers = 2
! Initial process time : 0.00s (0.00s extraction + 0.00s propagation)
! . Log search space : 664.4 (before), 664.4 (after)
! . Memory usage : 1.1 MB (before), 1.2 MB (after)
! Using parallel search with 2 workers.
! -----------------------------------------------------------------------
! Best Branches Non-fixed W Branch decision
* 1,062 739 0.01s 1 -
...
* 936 136k 3.69s 1 -
936 300k 85 2 F on op#0#9
! Time = 4.05s, Explored branches = 450,928, Memory usage = 7.7 MB
* 930 166k 4.55s 1 -
930 200k 110 1 F on op#5#1
! -----------------------------------------------------------------------
! Search terminated normally, 17 solutions found.
! Best objective : 930 (optimal - effective tol. is 0)
! Number of branches : 641,352
! Number of fails : 266,640
! Total memory usage : 6.9 MB (6.4 MB CP Optimizer + 0.5 MB Concert)
! Time spent in solve : 5.99s (5.99s engine + 0.00s extraction)
! Search speed (br. / s) : 107,070.5
! -----------------------------------------------------------------------
Problem
characteristics

Tips and Tools
! -----------------------------------------------------------------------
! Workers = 2
! -----------------------------------------------------------------------
* 1,062 739 0.01s 1 -
...
* 936 136k 3.69s 1 -
936 300k 85 2 F on op#0#9
* 930 166k 4.55s 1 -
930 200k 110 1 F on op#5#1
! -----------------------------------------------------------------------
! Search speed (br. / s) : 107,070.5
! -----------------------------------------------------------------------
Modified
parameter values

Tips and Tools
! -----------------------------------------------------------------------
! Workers = 2
! -----------------------------------------------------------------------
* 1,062 739 0.01s 1 -
...
* 936 136k 3.69s 1 -
936 300k 85 2 F on op#0#9
* 930 166k 4.55s 1 -
930 200k 110 1 F on op#5#1
! -----------------------------------------------------------------------
! Search speed (br. / s) : 107,070.5
! -----------------------------------------------------------------------
Root node
information

Tips and Tools
! -----------------------------------------------------------------------
! Workers = 2
! -----------------------------------------------------------------------
* 1,062 739 0.01s 1 -
...
* 936 136k 3.69s 1 -
936 300k 85 2 F on op#0#9
* 930 166k 4.55s 1 -
930 200k 110 1 F on op#5#1
! -----------------------------------------------------------------------
! Search speed (br. / s) : 107,070.5
! -----------------------------------------------------------------------
New incumbent
solutions (time, worker)

Tips and Tools
! -----------------------------------------------------------------------
! Workers = 2
! -----------------------------------------------------------------------
* 1,062 739 0.01s 1 -
...
* 936 136k 3.69s 1 -
936 300k 85 2 F on op#0#9
* 930 166k 4.55s 1 -
930 200k 110 1 F on op#5#1
! -----------------------------------------------------------------------
! Search speed (br. / s) : 107,070.5
! -----------------------------------------------------------------------
Periodical log
with fail information,
number of unfixed
variables, current decision

Tips and Tools
! -----------------------------------------------------------------------
! Workers = 2
! -----------------------------------------------------------------------
* 1,062 739 0.01s 1 -
...
* 936 136k 3.69s 1 -
936 300k 85 2 F on op#0#9
* 930 166k 4.55s 1 -
930 200k 110 1 F on op#5#1
! -----------------------------------------------------------------------
! Search speed (br. / s) : 107,070.5
! -----------------------------------------------------------------------
Final information
with solution status
and search statistics

Tips and Tools
 Search does not find any feasible solution, what should I do ?
–Identify which part of the model is hard:
• Simplify problem by removing “easy” constraints
• Search log helps (set LogPeriod=1, look where first fails occur)
–Strengthen the model:
• Be careful with composite constraints
• Add surrogate constraints
• Break symmetries
–Increase inference levels in the engine

Tips and Tools
 Search does not find any feasible solution, what should I do ?
–Try using search phases to first focus on the difficult decision
variables
–Solve the problem as two sub-problems:
1.Finding an initial feasible solution (some difficult constraints
can be relaxed and put in the objective function*
), then
2.Optimize starting from this solution. Consider using starting
points.
–Isolate the different sources of complexity in the model by
decomposition and sequential resolutions. Consider using
starting points
–(*) CP Optimizer's concepts makes it easy to relax part of the model.
E.g. make some interval variables optional and maximize number of
present intervals, relax deadlines as tardiness costs

Tips and Tools
 Search does not converge fast enough, what should I do ?
–Identify and try to isolate the different sources of
complexity in your model, first focusing on
bottleneck/important parts in a “simplified” model
–Exploit results of previous solve as (1) additional constraints
and/or (2) starting point and/or (3) heuristic (search phases)
–Do not hesitate to use MIP models in some steps, exploit
strengths of each technology (MIP/CP)
• For instance CP is good at deciding activities start/end times
under resource constraints when required resource quantities and
activity durations are fixed or mostly depend on activity start/end
dates
• For problems with very variable resource quantities or activity
durations, it makes sense to consider an initial MIP model that
works on these variables (example: lot sizing)

Tips and Tools
 Does my model improvement really help?
–CP Optimizer's search relies on a stochastic algorithm initialized
by a random seed (parameter IloCP::RandomSeed)
–For a given model, the solution depends on the random seed
–Best solution cost at a given time limit:
–
–
–
–
–
–
–
–
–Is model B better than model A ? Yes
Model A Model B
cost cost
seed=1
seed=1

Tips and Tools
 Does my model improvement really help?
–CP Optimizer's search relies on a stochastic algorithm initialized
by a random seed (parameter IloCP::RandomSeed)
–For a given model, the solution depends on the random seed
–Best solution cost at a given time limit:
–
–
–
–
–
–
–
–
–Is model B better than model A ? Yes
Model A Model B
cost cost
seed=1
seed=1
No!
Never evaluate a
model change on
a single instance/seed !

Conclusion
 For scheduling problems, CP Optimizer extends the Model&run
paradigm with some (few) general concepts (interval & sequence
variables, cumul & state functions, …)
 These concepts can be used to model scheduling problems in a
compact way that avoids enumeration of time. Typically, the
size of a scheduling model in CP Optimizer grows linearly with the
size of the data
 When combined together these concepts provide an expressive
language capable of handling complex and large industrial
scheduling problems
 The automatic search is complete, it exploits these concepts
using an efficient and robust combination of techniques from
scheduling theory, meta-heuristics, CP, MP

Agenda
 Tips and Tools
 Q&A

Bonus material
 List of most useful parameters for search analysis:
–Workers (=1)
–LogPeriod (=1)
–SearchType (=DepthFirst)
–FailLimit, SolutionLimit (=1)
 List of most useful parameters for performance tuning:
–TimeLimit
–TemporalRelaxation (=Off)
–Workers
–SearchType (=Restart|MultiPoint)
–Some inference levels:
• NoOverlapInferenceLevel
• CumulFunctionInferenceLevel

Bonus material: classical patterns in scheduling models
 Chain of optional interval variables
–Use cases: any phenomenon that is associated with a set of
consecutive time intervals whose number is not known a-priori
–Examples: partially preemptive activities, intervals during which
a resource is in use, …
 Model is a chain of precedence and implication constraints
between optional interval variables
endBeforeStart(a[i],a[i+1]);
presenceOf(a[i+1]) => presenceOf(a[i]);
 This ensures that only the first k intervals will be present where k
is an implicit decision of the problem
a1 a2 ai ai+1 an... ...

 Optional intervals in paths of precedence constraints
–Use case: a precedence graph with optional activities such that
path of precedence constraints going through absent activities
should still be considered
–Example: a → b c, precedence a c should be considered→ →
even if b is absent
–Direct model using endBeforeStart(a,b), endBeforeStart(b,c)
does not work as both constraints are inactive if b is absent
 Idea 1: Add endBeforeStart(x,y) constraints of the transitive
closure. Issue: may add a quadratic number of constraints
 Idea 2: Use additional non-optional interval variables A, B, C of
free length synchronized with a, b, c (startAtStart, endAtEnd)
a b c
A B C

 Multi-level alternatives
–Use case: an activity needs to select an element in a Cartesian
product
–Example: an activity A needs 1 worker Wi among n to be
executed at 1 given position Pj among m. Duration depends
both on the worker Wi and the position Pj. Specific constraints
need to be posted on all activities executing at a position Pj
(e.g. space limitation) while other constraints need to be
posted on all activities executed by a given worker Wi (e.g.
noOverlap)
 Idea: create one optional interval variable AW[i] for activity A
executing on worker Wi and one optional interval variable AP[j]
for activity A executing a position Pj

 Multi-level alternatives
 Idea: create one optional interval variable AW[i] for activity A
executing on worker Wi and one optional interval variable AP[j]
for activity A executing at position Pj
AW[i]
AW[n]
AW[1]
APW[n][1] APW[n][j] APW[n][m]
APW[i][1] APW[i][j] APW[i][m]
APW[1][1] APW[1][j] APW[1][m]
A AP[1] AP[j] AP[m]

 Temporal alternatives
–Alternative constraints are not restricted to model disjunctions
of resources/modes/recipes, they can also be used to
represent temporal disjunction
–Use-case: an activity must execute during a disjunctive set of
possible time-windows
 Alternative models that directly work on interval variable a are
possible (use cumulFunction or forbidExtent). They will be lighter
(less variables) but will propagate less.
a1 a2 a3
a
ALT

Bonus material
 Technical papers related with scheduling in CP Optimizer
– P. Vilím, P. Laborie and P. Shaw. “Failure-directed Search for Constraint-based Scheduling”. CP-AI-
OR 2015.
– P. Laborie. “An Optimal Iterative Algorithm for Extracting MUCs in a Black-box Constraint
Network”. ECAI 2014.
– P. Laborie and J. Rogerie. “Temporal linear relaxation in IBM ILOG CP Optimizer”. Journal of
Scheduling, Nov. 2014.
– P. Vilím. “Timetable Edge Finding Filtering Algorithm for Discrete Cumulative Resources”. CP-AI-OR
2011.
– P. Laborie, J. Rogerie, P. Shaw and P. Vilim. "Reasoning with Conditional Time-intervals - Part II:
an Algebraical Model for Resources". FLAIRS 2009.
– P. Laborie. "IBM ILOG CP Optimizer for Detailed Scheduling Illustrated on Three Problems". CP-AI-
OR 2009.
– P. Vilím. "Max Energy Filtering Algorithm for Discrete Cumulative Resources". CP-AI-OR 2009.
– P. Vilím. "Edge finding filtering algorithm for discrete cumulative resources in O(knlog(n))". CP
2009.
– P. Laborie and J. Rogerie. "Reasoning with Conditional Time-intervals". FLAIRS 2008.
– P. Laborie and D. Godard. "Self-Adapting Large Neighborhood Search: Application to single-mode
scheduling problems". MISTA 2007.
– P. Vilím. "Global Constraints in Scheduling". PhD thesis. Charles University, Prague. 2007.
– P. Laborie, D. Godard and W. Nuijten. “Randomized Large Neighborhood Search for Cumulative
Scheduling”. ICAPS 2005.

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