Integer Programming Approaches for Automated Planning. Menkes van den Briel Department of Industrial Engineering Arizona State University menkes@asu.edu http://www.public.asu.edu/~dbvan1/. What is automated planning?. Ordering problem
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Integer Programming Approaches for Automated Planning
Menkes van den BrielDepartment of Industrial EngineeringArizona State Universitymenkes@asu.eduhttp://www.public.asu.edu/~dbvan1/
Ordering problem
Scheduling is the problem of deciding when to execute a set of actions
NPcomplete
Selection and ordering problem
Planning is deciding both what actions need to be done and when to execute them
PSPACEcomplete
Scheduling
Planning
1
2
1
2
Initial states0 S
Goalg S
PlanP = a1, …, an
Action
Actions are state transformation functions
Initial states0 S
Goalg S
PlanP = a1, …, an
Action
si
sj
Actions are state transformation functions
Operations research (OR)
Scheduling problems typically involve solving hard optimization problems
Integer programming (IP), branchandbound
Artificial intelligence (AI)
Planning problems typically involve solving hard feasibility problems
Constraint satisfaction, satisfiability (SAT), A* search
Scheduling
Planning
[van den Briel, and Kambhampati, 2005]
1
2
P
T
Truck
Load(P,T,1)Unload(P,T,1)
1
Drive(1,2)
Drive(2,1)
2
Load(P,T,1)Unload(P,T,1)
Package
1
Load(P,T,1)
unload(P,T,1)
2
Load(P,T,2)
unload(P,T,2)
T
States are described by state variables
1
2
Prevail
Truck
Load(P,T,1)Unload(P,T,1)
1
Drive(1,2)
Drive(2,1)
2
Load(P,T,1)Unload(P,T,1)
Package
1
Load(P,T,1)
unload(P,T,1)
Effect
2
Load(P,T,2)
unload(P,T,2)
T
Actions are state transformation functions
Prevail
f
f
f
Effect
g
g
g
h
h
h
Plan step
Truck
1
1
2
2
Planning involves considering
plans of increasing length
Package
1
1
2
2
t
t
t = 1
Prevail
f
f
f
Effect
g
g
g
h
h
h
Load(P,T,1)
Drive(1,2)
Unload(P,T,2)
Truck
1
1
1
1
2
2
2
2
Load(P,T,1)

Unload(P,T,2)
Package
1
1
1
1
2
2
2
2
t
t
t
t
t = 1
t = 2
t = 3
Prevail
f
f
f
Effect
g
g
g
h
h
h
Load(P,T,1)Drive(1,2)
Unload(P,T,2)
Truck
1
1
1
2
2
2
Load(P,T,1)
Unload(P,T,2)
Package
1
1
1
2
2
2
t
t
t
t = 1
t = 2
A4
A1
A3
A1,A2
A3
A4
A2
Implied precendence graph
A4
A1
A3
A1,A2
A3
A4
A2
Implied precendence graph
A4
A1
xA1,t + xA3,t + xA4,t 2
aA:(f,f)SC(a)xa,t= ycf,g,tfor f, g Dc, f g,
xa,t ycf,f,t + gDc:f≠g (ycg,f,t + ycf,g,t)for a A, f PR(a)
aV()xa,t V() – 1for all cycles G, 1 t T
START
STOP
Initialize LP
no
Nodes found?
yes
LP solver
Feasible?
no
Fathom
Node selection
yes
Z_lp < Z*?
no
yes
Cut generation
Cuts found?
yes
no
Integer?
no
Branching
yes
Prevail
f
f
f
Effect
g
g
g
h
h
h
Load(P,T,1)Drive(1,2)Unload(P,T,2)
Truck
1
1
2
2
load(P,T,1)unload(P,T,2)
Package
1
1
2
2
t
t
t = 1
[van den Briel, Vossen, and Kambhampati, 2005, 2008]
Total Satisfaction Problems
PSP UTILITY COST
PSP NET BENEFIT
PLAN COST
PSP UTILITY
PSP GOAL LENGTH
Partial SatisfactionProblems
PLAN LENGTH
PSP GOAL
PLAN EXISTENCE
Plan step
Load(P,T,1)
Drive(1,2)
Unload(P,T,2)
Truck
1
1
1
1
2
2
2
2
Load(P,T,1)

Unload(P,T,2)
Package
1
1
1
1
2
2
2
2
t
t
t
t
t = 1
t = 2
t = 3
1
2
P
T
Truck
Load(P,T,1)Unload(P,T,1)
1
Drive(1,2)
Drive(2,1)
2
Load(P,T,1)Unload(P,T,1)
Package
1
Load(P,T,1)
unload(P,T,1)
2
Load(P,T,2)
unload(P,T,2)
T
No time indicesNo upper bounds
1if c c0,v, c g–1if c= c0,v, c g0otherwise
(1(c1,a), 2(c2,a)if a 1(c1)2(c2)(1(c1,a), c2)if a 1(c1)\A2(c1,2(c2,a))if a 2(c2)\A1undefinedotherwise
1
2
P
T
Truck
Load(P,T,1)Unload(P,T,1)
1
Drive(1,2)
Drive(2,1)
2
Load(P,T,1)Unload(P,T,1)
Package
1
Load(P,T,1)
unload(P,T,1)
2
Load(P,T,2)
unload(P,T,2)
T
1
2
P
T
Truck Package
2,1
Drive(2,1)
Drive(1,2)
1,2
1,1
Drive(2,1)
Drive(1,2)
Unload(P, T,1)
Load(P, T,1)
2,2
1,T
Unload(P, T,2)
Drive(2,1)
Load(P, T,2)
Drive(1,2)
2,T
Highlighted values equal optimal solution
BBOPLP planner
[Benton, van den Briel, and Kambhampati, 2007]
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