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Integrating Planning & Scheduling Subbarao Kambhampati
Integrating Planning & Scheduling
Agenda:
Questions on Scheduling?
Discussion on Smith’s paper?
Next topic: Nondeterministic Planning
Integrating Planning & Scheduling Subbarao Kambhampati
Dana Nau’s visit..
Talk on 3/31 afternoon (Monday)– Planning for Interactions among Autonomous Agents
May come to the class on 4/1 (Tuesday)– Can pick his brains…
Integrating Planning & Scheduling Subbarao Kambhampati
Temporal Planning & Scheduling
What we did: Action representations Search models
– Completeness considerations
Temporal networks Scheduling
Some outstanding issues Integrating planning &
scheduling Heuristics for temporal
planning
Integrating Planning & Scheduling Subbarao Kambhampati
Need for Integration Most existing planners
concentrate on action selection, ignoring resource allocation– Plan-based interfaces– Interactive decision support
Most existing schedulers concentrate only on resource allocation, ignoring action selection– E.g. HSTS operation
scheduling
Many real-world problems require both capabilities– Supply Chain Management problems
» I2, ILOG, Manugistics
– Planning in domains with durative actions, continuous change
» NASA RAX experiment
Integrating Planning & Scheduling Subbarao Kambhampati
Why now?
Significant scale-up in plan synthesis in last 4-5 years– 5/6 action plans in minutes to 100 action plans in
minutes – Breakthroughs in search space representation, heuristic
and domain-specific
Significant strides in our understanding of connections between planning and scheduling– Rich connections between planning and CSP/SAT/ILP
» Vanishing separation between planning techniques and scheduling techniques
Integrating Planning & Scheduling Subbarao Kambhampati
Approaches for Integration
Extend schedulers to handle action and resource choices
Extend planners to deal with resources, durative actions and continuous quantities
Coupled Architectures– De-coupled– Loosely Coupled (RealPlan System)
PLAN
NER
SCH
EDULE
RCausal Pla
n
Schdule
job-shopscheduling
planning
resourcechoices(RCSP)
alternativeprocesses
process3 process7process8
Task1
Task2
Task3
Task4
Task5
Task6
Task8
Task7
…
…… … …
… …
Integrating Planning & Scheduling Subbarao Kambhampati
Approaches
Decoupled– Existing approaches
Monolithic– Extend Planners to handle time and resources– Extend Schedulers to handle choice
Loosely Coupled– Making planners and schedulers interact
Integrating Planning & Scheduling Subbarao Kambhampati
Decoupled approaches(which is how Project Mgmt Done now)
Management
Technology Development
Mid-lower manager
Implementers
MS Project
(task planning)(scheduling)
Integrating Planning & Scheduling Subbarao Kambhampati
Extending Planners
ZENO [Penberthy & Weld], IxTET [Ghallab & Laborie], HSTS/RAX [Muscettola] extend a conjunctive plan-space planner with temporal and numeric constraint reasoners
LPSAT [Wolfman & Weld] integrates a disjunctive state-space planner with an LP solver to support numeric quantities
IPPlan [Kautz & Walser; 99] constructs ILP encodings with numeric constraints
TGP [Smith & Weld; 99] supports actions with durations in Graphplan
HSTSPlan Database
Model(DDL)
Searchengine
Engine
Goals
Initial state
Plan
Domain Knowledge
PlanningExperts
from EXEC
SearchControl
to EXEC
Integrating Planning & Scheduling Subbarao Kambhampati
Actions with Resources and Duration
Load(P:package, R:rocket, L:location)
Resources: ?h : robot hand Preconditions: Position(?h,L) [?s, ?e] Free(?h) ?s Charge(?h) > 5 ?s
Effects: holding(?h, P) [?s, ?t1] depositing(?h,P,R) [?t2, ?e] Busy(?h) [?s, ?e] Free(?h) ?e Charge - = .03*(?e - ?s) ?e
Constraints: ?t1 < ?t2 ?e - ?s in [1.0, 2.0]
Capacity(robot) = 3
?s ?e
Pos(?h,L)
Hold(?h,P)dep(?h,P)
[1,2]
Free(?h) Free(?h)Busy(?h)
Integrating Planning & Scheduling Subbarao Kambhampati
What planners are good for handling resources and time?
State-space approaches have an edge in terms of ease of monitoring resource usage– Time-point based representations are known to be better for multi-capacity
resource constraints in scheduling Plan-space approaches have an edge in terms of durative actions and continuous
change– Notion of state not well defined in such cases (Too many states)– PCP representations are known to be better for scheduling with single-capacity
resources
Integrating Planning & Scheduling Subbarao Kambhampati
Extending Scheduling
job-shopscheduling
planning
resourcechoices(RCSP)
alternativeprocesses
process3 process7process8
Task1
Task2
Task3
Task4
Task5
Task6
Task8
Task7
Ordering choicesResource choicesProcess choices
Integrating Planning & Scheduling Subbarao Kambhampati
Monolithic Architectures Scale Poorly
Extended planning systems are hard to control
– RAX uses a very error-prone hand-coded search control strategy
Extended scheduling systems tend to lose effectiveness due to increased disjunction
Monolithic systems can sometimes show counter-intuitive behavior (by multiplying search failures)
D
C
E
B
F
A
E
B
A
F
D
C
Total
Search
Graph
Performance worsens with increased resources!
Integrating Planning & Scheduling Subbarao Kambhampati
Loosely Coupled Architectures
Schedulers already routinely handle resources and metric/temporal constraints. – Let the “planner”concentrate on causal reasoning– Let the “scheduler” concentrate on resource
allocation, sequencing and numeric constraints for the generated causal plan
PLANNER
SCHEDULE
R
Cau
sal P
lan
Schd
ule
Need better coupling to avoid inter-module thrashing….
Integrating Planning & Scheduling Subbarao Kambhampati
Making Loose Coupling Work
How can the Planner keep track of consistency?
– Low level constraint propagation
» Loose path consistency on TCSPs
» Bounds on resource consumption,
» LP relaxations of metric constraints
– Pre-emptive conflict resolution
The more aggressive you do this, the less need for a scheduler.. How do the modules interact?
– Failure explanations; Partial results
PLAN
NER
SCH
EDULE
RCausal Pla
n
Schdule
Integrating Planning & Scheduling Subbarao Kambhampati
RealPlan--Master/Slave
PS
Master-Slave(RealPlan-MS)
Variables: ActionsValues: Resources
Planner does causal reasoning.
Scheduler attempts resource allocation If scheduler fails, planner has to restart
[Sriv
astava
&
Kam
bhampati
ECP,9
9;
AAAI,
2000]
Planner’s CSP: Variables: “goals” Values: “actions”
Scheduler’s CSP: Variables: “Actions” Values: “Resources”
Resou
rce c
onstr
aint
s
activ
ated
by
the
sele
cted
acto
ns
Integrating Planning & Scheduling Subbarao Kambhampati
Performance of Master-Slave Coupling
When sc
heduler f
ails, n
o
specifi
c guid
ance is
given
to th
e planner
Integrating Planning & Scheduling Subbarao Kambhampati
RealPlan: Peer-to-Peer
SP T
Peer-Peer (RealPlan-PP)
Variables: FactsValues: Actions
Variables: ActionsValues: Resources
Explanation-dire
cted
backtracking b
etween
Planner and Scheduler
Planner’s CSP: Variables: “goals” Values: “actions”
Scheduler’s CSP: Variables: “Actions” Values: “Resources”
Set o
f act
ions
that
cann
ot b
e sch
edul
ed
Resou
rce c
onstr
aint
s
activ
ated
by
the
sele
cted
acto
ns
Integrating Planning & Scheduling Subbarao Kambhampati
Inter-module Dependency Directed Backtracking
Explanation Generation
Explanation Translation
Generation of Alternative Plan
Scheduler’s Task
Interface’s Task
Planner’s Task
Generate compact explanation of theScheduler’s failure in allocating resources
Translate the explanation into the formthat make sense to the planner.
Use the translated explanation togenerate plan that avoid this failure.
Planner’s CSP:Variables: “goals”Values: “actions”
Scheduler’s CSP:Variables: “Actions”Values: “Resources”
Set o
f act
ions
that
cann
ot b
e sch
edul
ed
Resou
rce c
onstr
aint
s
activ
ated
by
the
sele
cted
act
ons
Integrating Planning & Scheduling Subbarao Kambhampati
Resource Domains:A1, A2 , A3: {R1 , R2}A4, A5: {S1 , S2 , S3}
Resource Constraints:A1 A2 ; A2 A3 ;A1 A3 ; A4 A5 ;
N1: {A1 = R1}
N1: {A1 = R1, A2 = R2}
N1: {A1 = R1, A2 = R2 , A4 = S1}
N1: {A1 = R1, A2 = R2 , A4 = S1 , A3 = R1} N1: {A1 = R1, A2 = R2 , A4 = S1 , A3 = R2}
A1 = R1
A2 = R2
A4 = S1
A3 = R1 A3 = R2
Subset of variables that can not beassigned values (reason of failure):
(A1 , A2 , A3)
Integrating Planning & Scheduling Subbarao Kambhampati
Shuffle 8 Blocks
1
10
100
1000
10000
100000
2 3 4 5
Robot hand
Tim
e (s
ec)
Realplan-PPRealplan-MS
RealPlan-PP RealPlan-MS Speedup (x)ProblemClass Time #bk
BTPWFTime Class Time BTPWF RealP
lan-MS
Shuf_b8r4
Shuf_b8r4
INFRES
FIX
2.52
8.74
4
1
>3 hr
>3 hr
SAMELEN
SAMELEN
9373
9373
>4170
>1236
3619
1072
Huge_r6 INFRES 4.53 4 976 INCRLEN 11056 215 2441
Huge_r6
Huge_r6
FIX
SAMELEN
17.48
20.59
3
2
991
983
INCRLEN
INCRLEN
11056
11056
56.7
47.7
632
537
Integrating Planning & Scheduling Subbarao Kambhampati
A temporal plannersupporting causal plan synthesis
CSP-basedFinite capacityresource scheduler
Mixed Integer/Linearprogramming modulefor metric constraints
Mission Profile
Executor
inte
rfac
e
interface
interface
Plans with annotatedwaypoints
Execution statusReplanning requests
Mission modificationsPlan criticism
Evolving specs
Next-Generation Realplan
Integrating Planning & Scheduling Subbarao Kambhampati
Summary & Conclusion
Motivated the need for integrating Planning and Scheduling
Discussed the state of the art in Planning and Scheduling
Discussed approaches for Integrating them– Loosely coupled architectures are a promising
approach
26
Multi-objective search
Multi-dimensional nature of plan quality in metric temporal planning: Temporal quality (e.g. makespan, slack—the time when a
goal is needed – time when it is achieved.) Plan cost (e.g. cumulative action cost, resource consumption)
Necessitates multi-objective optimization: Modeling objective functions Tracking different quality metrics and heuristic estimation Challenge: There may be inter-dependent
relations between different quality metric
27
Example
Option 1: Tempe Phoenix (Bus) Los Angeles (Airplane) Less time: 3 hours; More expensive: $200
Option 2: Tempe Los Angeles (Car) More time: 12 hours; Less expensive: $50
Given a deadline constraint (6 hours) Only option 1 is viable Given a money constraint ($100) Only option 2 is viable
Tempe
Phoenix
Los Angeles
28
Solution Quality in the presence of multiple objectives
When we have multiple objectives, it is not clear how to define global optimum
E.g. How does <cost:5,Makespan:7> plan compare to <cost:4,Makespan:9>? Problem: We don’t know what the user’s utility metric
is as a function of cost and makespan.
29
Solution 1: Pareto Sets
Present pareto sets/curves to the user A pareto set is a set of non-dominated solutions
A solution S1 is dominated by another S2, if S1 is worse than S2 in at least one objective and equal in all or worse in all other objectives. E.g. <C:4,M9> dominated by <C:5;M:9>
A travel agent shouldn’t bother asking whether I would like a flight that starts at 6pm and reaches at 9pm, and cost 100$ or another ones which also leaves at 6 and reaches at 9, but costs 200$.
A pareto set is exhaustive if it contains all non-dominated solutions Presenting the pareto set allows the users to state their preferences implicitly by
choosing what they like rather than by stating them explicitly. Problem: Exhaustive Pareto sets can be large (exponentially large in many cases).
In practice, travel agents give you non-exhaustive pareto sets, just so you have the illusion of choice
Optimizing with pareto sets changes the nature of the problem—you are looking for multiple rather than a single solution.
30
Solution 2: Aggregate Utility Metrics Combine the various objectives into a single utility measure
Eg: w1*cost+w2*make-span Could model grad students’ preferences; with w1=infinity, w2=0
Log(cost)+ 5*(Make-span)25 Could model Bill Gates’ preferences.
How do we assess the form of the utility measure (linear? Nonlinear?) and how will we get the weights?
Utility elicitation process Learning problem: Ask tons of questions to the users and learn their utility function to fit their
preferences Can be cast as a sort of learning task (e.g. learn a neual net that is consistent with the examples)
Of course, if you want to learn a true nonlinear preference function, you will need many many more examples, and the training takes much longer.
With aggregate utility metrics, the multi-obj optimization is, in theory, reduces to a single objective optimization problem *However* if you are trying to good heuristics to direct the search, then since estimators are
likely to be available for naturally occurring factors of the solution quality, rather than random combinations there-of, we still have to follow a two step process
1. Find estimators for each of the factors2. Combine the estimates using the utility measure THIS IS WHAT IS DONE IN SAPA
31
Sketch of how to get cost and time estimates
Planning graph provides “level” estimates Generalizing planning graph to “temporal planning graph” will allow us to
get “time” estimates For relaxed PG, the generalization is quite simple—just use bi-level
representation of the PG, and index each action and literal by the first time point (not level) at which they can be first introduced into the PG
Generalizing planning graph to “cost planning graph” (i.e. propagate cost information over PG) will get us cost estimates
We discussed how to do cost propagation over classical PGs. Costs of literals can be represented as monotonically reducing step functions w.r.t. levels.
To estimate cost and time together we need to generalize classical PG into Temporal and Cost-sensitive PG
Now, the costs of literals will be monotonically reducing step functions w.r.t. time points (rather than level indices)
This is what SAPA does
32
SAPA approach
Using the Temporal Planning Graph (Smith & Weld) structure to track the time-sensitive cost function: Estimation of the earliest time (makespan) to achieve all goals. Estimation of the lowest cost to achieve goals Estimation of the cost to achieve goals given the specific
makespan value. Using this information to calculate the heuristic
value for the objective function involving both time and cost
Involves propagating cost over planning graphs..
33
Heuristics in Sapa are derived from the Graphplan-stylebi-level relaxed temporal planning graph (RTPG)
Progression; so constructed anew for each state..
34
Relaxed Temporal Planning Graph
Relaxed Action:No delete effects
May be okay given progression planningNo resource consumption
Will adjust later
PersonAirplane
Person
A B
Load(P,A)
Fly(A,B) Fly(B,A)
Unload(P,A)
Unload(P,B)
Init Goal Deadline
t=0 tg
while(true) forall Aadvance-time applicable in S S = Apply(A,S)
Involves changing P,,Q,t{Update Q only with positive effects; and only when there is no other earlier event giving that effect}
if SG then Terminate{solution}
S’ = Apply(advance-time,S) if (pi,ti) G such that ti < Time(S’) and piS then Terminate{non-solution} else S = S’end while; Deadline goals
RTPG is modeled as a time-stamped plan! (but Q only has +ve events)
Note: Bi-level rep; we don’t actually stack actions multiple times in PG—we just keep track the first time the action entered
36
Heuristics directly from RTPG
For Makespan: Distance from a state S to the goals is equal to the duration between time(S) and the time the last goal appears in the RTPG.
For Min/Max/Sum Slack: Distance from a state to the goals is equal to the minimum, maximum, or summation of slack estimates for all individual goals using the RTPG. Slack estimate is the difference
between the deadline of the goal, and the expected time of achievement of that goal.
Proof: All goals appear in the RTPG at times smalleror equal to their achievable times.
ADMISSIBLE
PersonAirplane
Person
A B
Load(P,A)
Fly(A,B) Fly(B,A)
Unload(P,A)
Unload(P,B)
Init Goal Deadline
t=0 tg
PersonAirplane
Person
A B
Load(P,A)
Fly(A,B) Fly(B,A)
Unload(P,A)
Unload(P,B)
Init Goal Deadline
t=0 tg
37
Heuristics from Relaxed Plan Extracted from RTPG
RTPG can be used to find a relaxed solution which is thenused to estimate distance from a given state to the goals
Sum actions: Distance from a state S to the goals equals the number of actions in the relaxed plan.
Sum durations: Distance from a state S to the goals equals the summation of action durations in the relaxed plan.
PersonAirplane
Person
A B
Load(P,A)
Fly(A,B) Fly(B,A)
Unload(P,A)
Unload(P,B)
Init Goal Deadline
t=0 tg
38
Resource-based Adjustments to Heuristics
Resource related information, ignored originally, can be used to improve the heuristic values
Adjusted Sum-Action:
h = h + R (Con(R) – (Init(R)+Pro(R)))/R
Adjusted Sum-Duration:
h = h + R [(Con(R) – (Init(R)+Pro(R)))/R].Dur(AR)
Will not preserve admissibility
39
The (Relaxed) Temporal PG
Tempe
Phoenix
Los Angeles
Drive-car(Tempe,LA)
Heli(T,P)
Shuttle(T,P)
Airplane(P,LA)
t = 0 t = 0.5 t = 1 t = 1.5 t = 10
40
Time-sensitive Cost Function
Standard (Temporal) planning graph (TPG) shows the time-related estimates e.g. earliest time to achieve fact, or to execute action
TPG does not show the cost estimates to achieve facts or execute actions
Tempe
Phoenix
L.A
Shuttle(Tempe,Phx): Cost: $20; Time: 1.0 hourHelicopter(Tempe,Phx):Cost: $100; Time: 0.5 hourCar(Tempe,LA):Cost: $100; Time: 10 hourAirplane(Phx,LA):Cost: $200; Time: 1.0 hour
cost
time0 1.5 2 10
$300
$220
$100
Drive-car(Tempe,LA)
Heli(T,P)
Shuttle(T,P)
Airplane(P,LA)
t = 0 t = 0.5 t = 1 t = 1.5 t = 10
41
Estimating the Cost Function
Tempe
Phoenix
L.A
time0 1.5 2 10
$300
$220
$100
t = 1.5 t = 10
Shuttle(Tempe,Phx): Cost: $20; Time: 1.0 hourHelicopter(Tempe,Phx):Cost: $100; Time: 0.5 hourCar(Tempe,LA):Cost: $100; Time: 10 hourAirplane(Phx,LA):Cost: $200; Time: 1.0 hour
1
Drive-car(Tempe,LA)
Hel(T,P)
Shuttle(T,P)
t = 0
Airplane(P,LA)
t = 0.5
0.5
t = 1
Cost(At(LA)) Cost(At(Phx)) = Cost(Flight(Phx,LA))
Airplane(P,LA)
t = 2.0
$20
42
Observations about cost functions
Because cost-functions decrease monotonically, we know that the cheapest cost is always at t_infinity (don’t need to look at other times) Cost functions will be monotonically decreasing as long as there are no exogenous
events Actions with time-sensitive preconditions are in essence dependent on exogenous
events (which is why PDDL 2.1 doesn’t allow you to say that the precondition must be true at an absolute time point—only a time point relative to the beginning of the action
If you have to model an action such as “Take Flight” such that it can only be done with valid flights that are pre-scheduled (e.g. 9:40AM, 11:30AM, 3:15PM etc), we can model it by having a precondition “Have-flight” which is asserted at 9:40AM, 11:30AM and 3:15PM using timed initial literals)
Becase cost-functions are step funtions, we need to evaluate the utility function U(makespan,cost) only at a finite number of time points (no matter how complex the U(.) function is. Cost functions will be step functions as long as the actions do not model
continuous change (which will come in at PDDL 2.1 Level 4). If you have continuous change, then the cost functions may change continuously too
ADDED
44
Cost Propagation Issues:
At a given time point, each fact is supported by multiple actions Each action has more than one precondition
Propagation rules: Cost(f,t) = min {Cost(A,t) : f Effect(A)} Cost(A,t) = Aggregate(Cost(f,t): f Pre(A))
Sum-propagation: Cost(f,t) The plans for individual preconds may be interacting
Max-propagation: Max {Cost(f,t)} Combination: 0.5 Cost(f,t) + 0.5 Max {Cost(f,t)}
Probably other better ideas could be tried
Can’t use something like set-level idea here becauseThat will entail tracking the costs of subsets of literals
45
Termination Criteria
Deadline Termination: Terminate at time point t if: goal G: Dealine(G) t goal G: (Dealine(G) < t) (Cost(G,t) =
Fix-point Termination: Terminate at time point t where we can not improve the cost of any proposition.
K-lookahead approximation: At t where Cost(g,t) < , repeat the process of applying (set) of actions that can improve the cost functions k times.
cost
time0 1.5 2 10
$300
$220
$100
Drive-car(Tempe,LA)
H(T,P)
Shuttle(T,P)
Plane(P,LA)
t = 0 0.5 1 1.5 t = 10
Earliest time pointCheapest cost
46
Heuristic estimation using the cost functions
If the objective function is to minimize time: h = t0
If the objective function is to minimize cost: h = CostAggregate(G, t)
If the objective function is the function of both time and cost
O = f(time,cost) then:h = min f(t,Cost(G,t)) s.t. t0 t t
Eg: f(time,cost) = 100.makespan + Cost then h = 100x2 + 220 at t0 t = 2 t
time
cost
0 t0=1.5 2 t = 10
$300
$220
$100
Cost(At(LA))
Earliest achieve time: t0 = 1.5Lowest cost time: t = 10
The cost functions have information to track both temporal and costmetric of the plan, and their inter-dependent relations !!!
47
Heuristic estimation by extracting the relaxed plan
Relaxed plan satisfies all the goals ignoring the negative interaction: Take into account positive interaction Base set of actions for possible adjustment according to
neglected (relaxed) information (e.g. negative interaction, resource usage etc.)
Need to find a good relaxed plan (among multiple ones) according to the objective function
49
Heuristic estimation by extracting the relaxed plan
General Alg.: Traverse backward searching for actions supporting all the goals. When A is added to the relaxed plan RP, then:
Supported Fact = SF Effects(A)Goals = SF \ (G Precond(A))
Temporal Planning with Cost: If the objective function is f(time,cost), then A is selected such that:
f(t(RP+A),C(RP+A)) + f(t(Gnew),C(Gnew)) is minimal (Gnew = (G Precond(A)) \ Effects)
Finally, using mutex to set orders between A and actions in RP so that less number of causal constraints are violated
time
cost
0 t0=1.5 2 t = 10
$300
$220
$100
Tempe
Phoenix
L.A
f(t,c) = 100.makespan + Cost
50
Adjusting the Heuristic Values
Ignored resource related information can be used to improve the heuristic values (such like +ve and –ve interactions in classical planning)
Adjusted Cost:
C = C + R (Con(R) – (Init(R)+Pro(R)))/R * C(AR)
Cannot be applied to admissible heuristics
51
Partialization Example
A1 A2 A3
A1(10) gives g1 but deletes pA3(8) gives g2 but requires p at startA2(4) gives p at end We want g1,g2
A position-constrained plan with makespan 22
A1
A2
A3 G
p
g1
g2
[et(A1) <= et(A2)] or [st(A1) >= st(A3)][et(A2) <= st(A3)….
OrderConstrainedplan
The best makespan dispatch of the order-constrained plan
A1
A2 A3 14+
There could be multiple O.C. plansbecause of multiple possible causal sources. Optimization will involve Going through them all.
52
Problem Definitions Position constrained (p.c) plan: The execution time of each action is
fixed to a specific time point Can be generated more efficiently by state-space planners
Order constrained (o.c) plan: Only the relative orderings between actions are specified More flexible solutions, causal relations between actions
Partialization: Constructing a o.c plan from a p.c plan
QR R
G
QR
{Q} {G}
t1 t2 t3
p.c plan o.c plan
Q R RG
QR
{Q} {G}
53
Validity Requirements for a partialization
An o.c plan Poc is a valid partialization of a valid p.c plan Ppc, if: Poc contains the same actions as Ppc
Poc is executable Poc satisfies all the top level goals (Optional) Ppc is a legal dispatch (execution) of Poc
(Optional) Contains no redundant ordering relations
PQ
PQ
Xredundant
54
Greedy Approximations
Solving the optimization problem for makespan and number of orderings is NP-hard (Backstrom,1998)
Greedy approaches have been considered in classical planning (e.g. [Kambhampati & Kedar, 1993], [Veloso et. al.,1990]):
Find a causal explanation of correctness for the p.c plan Introduce just the orderings needed for the explanation to
hold
56
Modeling greedy approaches as value ordering strategies
Variation of [Kambhampati & Kedar,1993] greedy algorithm for temporal planning as value ordering: Supporting variables: Sp
A = A’ such that: etp
A’ < stpA in the p.c plan Ppc
B s.t.: etpA’ < etp
B < stpA
C s.t.: etpC < etp
A’ and satisfy two above conditions Ordering and interference variables:
pAB = < if etp
B < stpA ; p
AB = > if stpB > stp
A
rAA’= < if etr
A < strA’ in Ppc; r
AA’= > if strA > etr
A’ in Ppc; rAA’= other wise.
Key insight: We can capture many of the greedy approaches as specific value ordering strategies on the CSOP encoding
60
Empirical evaluation
Objective: Demonstrate that metric temporal planner armed with our
approach is able to produce plans that satisfy a variety of cost/makespan tradeoff.
Testing problems: Randomly generated logistics problems from TP4
(Hasslum&Geffner)Load/unload(package,location): Cost = 1; Duration = 1;Drive-inter-city(location1,location2): Cost = 4.0; Duration = 12.0;Flight(airport1,airport2): Cost = 15.0; Duration = 3.0;Drive-intra-city(location1,location2,city): Cost = 2.0; Duration = 2.0;