Planning with Resources

Planning with Resources

Philippe [email protected]

2 PLANET International Summer School on AI Planning, Sept. 2002, Halkidiki, Greece

Applications of AI Planning

AerospaceSpace

- Satellite, Launcher manufacturing

- Launch site operations

- Spacecraft operations

- Space users

- Autonomous space exploration

Aviation

- Manufacturers and suppliers

- Airlines, Airports

- Maintenance and repair organizations

- Air Traffic Control

Foreword



Defense

Military logistic applications

Rescue and Military operations planning

Non-combatant Evacuation Operations

Emergency Assistance

Air Campaign Planning

Army Unit Operations

Coalition Operations

Foreword



New information technology

Planning & Scheduling for the Web

Workflow, Business Process Management

Systems Management Aids

Help Desks and Assistants

Personal Assistants

Data analysis tasks, Database query

Virtual reality or other simulated environments

Computer games (bridge)

Foreword



Industry

Process planning

Factory automation

- Production line scheduling

- Assembly planning

- Maintenance planning

Electronic design and manufacturing

Logistics

Project planning

Construction planning

Engineering Tasks

Management Tasks

Foreword



Civil Security and Environment Crisis Management

Evacuation Planning

Search & Rescue Coordination

Oil Spill Response

Medical ApplicationsBioinformatics

Personal Active Medical Assistant

Foreword



What most of these applications have in common ?

Foreword

Manufacturing: A grinding operation requires a grinding machine. The grinding machine can perform only one operation at a time. Only one grinding machine is available in the shop.

Project Management: In a web site development project, the task of testing browsers compatibility requires an effort of 6 person-months of a quality engineer. 5 quality engineers are available.

Satellite: When the satellite is not within range of a receiving station, it can store images on an onboard solid-state memory that provides a capacity of 90 Gbits (550 images).

Crisis Management: An aircraft B737-200 has a capacity to evacuate at most 120 people. Maximum cargo loading is 3,400kg for a volume of 14.3 m3. Maximal range of the aircraft is 2480km.

… they need the ability to reason about resources



Part I. State of the Art Overview Definition of a Resource Problem Modeling Problem Solving

Part II. Focus on Constraint-Based Approaches Constraint-Programming Problem Modeling Problem Solving

Constraint Propagation Search

Overview


Part I: State of the Art Overview

10

PLANET International Summer School on AI Planning, Sept. 2002, Halkidiki, Greece

Definition of a Resource

A resource is any substance or (set of) object(s) whose cost or available quantity induce some constraint on the operations that use it

State of the Art

Manufacturing: A grinding operation requires a grinding machine. The grinding machine can perform only one operation at a time. Only one grinding machine is available in the shop.

The grinding machine is a resource. It constrains the grinding operations to execute sequentially on the machine.

0

1

Grinding-op1 Grinding-op3 Grinding-op2 time

11




State of the Art

The team of quality engineers is a resource. It constrains the number of tasks that can be executed in parallel and the duration of these tasks.

Project Management: In a web site development project, the task of testing browsers compatibility requires an effort of 6 person-months of a quality engineer. 5 quality engineers are available.

0

time2 engineers

3 months

Web-site-dev1

Test-software1

5

12




State of the Art

The solid-state memory is a resource. It constrains the number of images that can be taken between two periods of visibility of the receiving station.

Satellite: When the satellite is not within range of a receiving station, it can store images on an onboard solid-state memory that provides a capacity of 90 Gbits (550 images).

0

550

timeSend-dataPic1 Pic2 Pic5 Pic3 Pic4

13




State of the Art

The seat capacity of the B737-20 is a resource. It constrains the number of people the plane can evacuate in a single flight. The cargo capacity of the B737-20 is another resource. It constrains the weight the aircraft can transport in a single flight. Etc...

Crisis Management: An aircraft B737-200 has a capacity to evacuate at most 120 people. Maximum cargo loading is 3,400kg for a volume of 14.3 m3. Maximal range of the aircraft is 2480km.

0

3400

timeLoad-truck Load-car Unload-car Unload-truck

14


State of the Art


Resources Typology

Unary resource: Resource of capacity 1. Two activities requiring the same unary resource cannot overlap

Discrete resource: Resource of capacity Q. Activities requiring the same discrete resource can overlap provided the resource capacity Q is not exceeded

Reservoir: Resource of capacity Q and initial level L; Activities may consume or produce the reservoir. The level of the reservoir over time must be kept in the interval [0,Q]

15


State of the Art


unary resource availability

0

1

timeOp1 Op2

16


State of the Art


discrete resource availability

0

Q

timeOp1

Op2

17


State of the Art


In theory, a discrete resource of maximal capacity Q can be represented as Q unary resources

Op1 {R1,…,Ri,…,Rj,…,RQ}

Opi {R1,…,Ri,…,Rj,…,RQ}

Opj {R1,…,Ri,…,Rj,…,RQ}

OpQ {R1,…,Ri,…,Rj,…,RQ}

OpQ+1 {R1,…,Ri,…,Rj,…,RQ}

No solution after trying Q! allocations

18


State of the Art


reservoir level

0

Q

timeOp1

Op2

Op3

19


State of the Art


Resources are tightly linked with the notions of time & concurrency For unary and discrete resources, activities use the

resource over some time interval Handling parallel execution of activities is necessary to

take advantage of non-unitary capacities There may exist complex relations linking the duration

of an activity with the amount of resource it uses, consumes and/or produces

20


State of the Art


Resources may impose other constraints than just capacity State constraints : two activities requiring the resource

to be in incompatible states cannot overlap Minimal transition time between two consecutive

activities using the same unary resource Batching: two overlapping activities using the same

resource must start and end at the same time These additional constraints can generally be

expressed by classical planning predicates

21


State of the Art


So … the hard point for planning with resources is mainly the management of Numerical time and Numerical capacity/availability

22


Problem Modeling: STRIPS

How hard is it to model time and capacities in STRIPS ? Exercise. Build a STRIPS model for the following

domain

A task to test a web-site lasts for 3 months and requires 2 quality engineers (out of 5 available). As a result of the task, the web-site is tested. Other tasks also have duration and compete for the same resource. The model must correctly handle task parallelism.

State of the Art

23


Problem Modeling : STRIPS

Difficulty #1: Numerical quantities No integer arithmetic is available in STRIPS. We need

to provide one. In domain description

(:predicates (plus ?i ?j ?k))

In problem definition

(:objects _0 _1 _2 _3 _4 _5 _6 _7 _8 … )

(:init (plus _0 _1 _1)(plus _1 _1 _2)(plus _2 _1 _3) …

(plus _0 _2 _2)(plus _1 _2 _3)(plus _2 _2 _4) … )

State of the Art

24



Difficulty #1: Numerical quantities No integer arithmetic is available in STRIPS. We need

to provide one. Example of action that increase a quantity

(:action IncreaseQuantity :parameters (?i ?d ?j) :precondition (and (quantity ?i) (plus ?i ?d ?j)) ; ?j == ?i + ?d :effect (and (not (quantity ?i))

(quantity ?j)))

State of the Art

25



Difficulty #2: Discrete Resource The task is a time interval over which the resource is

used. Two actions to represent the task

(:action TestWebSite-Start :parameters (?q2 ?q0) :precondition (and (free-engineers ?q2) (plus ?q0 _2 ?q2))

:effect (and (not (free-engineers ?q2)) (free-engineers ?q0)))

State of the Art

26




used. Two actions to represent the task

(:action TestWebSite-End :parameters (?q0 ?q2) :precondition (and (free-engineers ?q0) (plus ?q0 _2 ?q2)) :effect (and (not (free-engineers ?q0))

(free-engineers ?q2)(web-site-tested)))

State of the Art

27




used. Initially, the amount of resource is 5:

(:init (free-engineers _5) … )

State of the Art

28



Difficulty #3: Task Duration Use a predicate startedAtTask to record when the

task was started and a global clock clock to measure time

(:predicates (clock ?t)(testwebsite_started_at ?t) … )

(:action TestWebSite-Start :parameters (?t0 … ) :precondition (and (clock ?t0) … )

:effect (and (testwebsite_started_at ?t0) … ))

State of the Art

29



Difficulty #3: Task Duration

(:action TestWebSite-End :parameters (?t3 ?t0 … ) :precondition (and (testwebsite_started_at ?t0)

(clock ?t3)(plus ?t0 _3 ?t3) … )

:effect (and (not (testwebsite_started_at ?t0) … )))

State of the Art

30



Difficulty #3: Task Duration Now we only need an action to increase time when

necessary:

(:action Tick :parameters (?t0 ?t1) :precondition (and (clock ?t0)

(plus ?t0 _1 t1)) :effect (and (not (clock ?t0))

(clock ?t1)))

We start at time 0:

(:init (clock I0) … )

State of the Art

31



Conclusion It is possible to represent time and resources in STRIPS BUT … the model is not completely intuitive

No built-in arithmetic A relative change on a resource (free-engineers -= 2) is

represented by two absolute changes:(DEL (free-engineers ?x)) (ADD (free-engineers ?x+2))

Representation of time is not natural (action TICK)

State of the Art

32



Conclusion A relative change on a resource is represented by two

absolute changes

State of the Art

(PRECOND (free-engineers ?x1)(plus ?x1 _1 ?x2)) (DEL (free-engineers ?x1)) (ADD (free-engineers ?x2))

(PRECOND (free-engineers ?y1)(plus ?y1 _1 ?y2)) (DEL (free-engineers ?y1)) (ADD (free-engineers ?y2))

MUTEX

33



Conclusion AND what about the performances ?

State of the Art

150s580s

26s

TestWebSiteTest

SoftwareTraining

engineers

TestWebSite

TestSoftware

Training

34


Problem Modeling

Beyond STRIPS

Standalone languages

PDDL2.1 BTPL

Planning systems (language)

parcPLAN O-Plan

IXTET ASPEN

RAX-PS

State of the Art

35


State of the Art

Problem Modeling : PDDL 2.1

PDDL 2.1 was developed for the AIPS-2002 Planning Competition [Fox&Long 2002]

Extension of PDDL [McDermott&al 1998] for domains involving time and other numeric-valued resources

PDDL 2.1 Level 1: “Classical” propositional planning Level 2: Numeric Variables Level 3: Durative actions, No continuous effects Level 4: Durative actions, continuous effects

36


State of the Art


Numeric variables

(:functions (free-engineers) )

(:action Hire :parameters ( ?n ) :precondition ( … )

:effect (and (increase (free-engineers) ?n) … ))

37


State of the Art


Numeric variables

Conditions:(= (f ?x) ?q)(< (f ?x) ?q) (> (f ?x) ?q)(<= (f ?x) ?q) (>= (f ?x) ?q)

Effects:(increase (f ?x) ?q) ; f(?x) += ?q(decrease (f ?x) ?q) ; f(?x) -= ?q (scale-up (f ?x) ?q) ; f(?x) *= ?q(scale-down (f ?x) ?q) ; f(?x) /= ?q(assign (f ?x) ?q) ; f(?x) = ?q

38


State of the Art


Durative actions

Action

Start Endduration

Conditionsat end

Effectsat end

Conditionsat start

Effectsat start

Invariant

39


State of the Art


Durative actions

(:functions (free-engineers) )

(:action TestWebSite :parameters ( )

:duration (= ?duration 3) :condition (and (at start (>= (free-engineers) 2)))

:effect (and (at start (decrease (free-engineers) 2))

(at end (increase (free-engineers) 2))

(at end (website_tested))))

40


State of the Art

Problem Modeling : BTPL

Basic Temporal Planning Language [Brenner 2001] Extension of PDDL. Focus on time and concurrency. Whereas PDDL2.1 represents an action with two events

(start and end), BTPL allows for more complex actions involving several events

No relative changes on numerical attributes (increase/decrease) as in PDDL2.1

41


State of the Art


A BTPL ground action is a tuple a=(pre, eff) where pre are preconditions (set of literals)

eff = {ue1,..,uen} is a set of unit events of the form ue=(c,e,d) describing the (conditional) effects of A d: delay (rational number) c: condition of ue: set of literals to be satisfied at t+d e: effect of ue: add/delete list occurring at t+d

Apre

ue1

c1 e1d1

ue2

c2 e2d2

42


State of the Art


Example: ATM cash dispenser

DialCode(?x)(card_in ?y)

(code_ok ?x ?y)

5

(ticket)7

(cash-out)

10not (cash-out) not(card_in ?y)

TakeCash()(cash-out)

not(cash-out)(have_cash)1

InsertCard(?y)

(card_in ?y)

1

43


State of the Art


BTPL extension Allow expressing action preconditions (pre) and event

conditions (c) that hold over time intervals [t1,t2) before the action or event time-point.

pre: (p,d1,d2)

t-d2t-d1

A

ue ed

t

t’=t+d

c: (q,d3,d4)

t’-d4t’-d3

44


State of the Art

Problem Modeling : parcPLAN

parcPLAN [Lever&Richards 1994, ElKholy&Richards 1996] Rich temporal expressivity based on time-intervals

whose extremities are time-point variables

E.g.: position(robot,r1)@(t1,t2) Expression of minimal/maximal delays between time

points

E.g.: t1+5 t2 Handle unary and discrete resources, no reservoir

45


State of the Art


Web-site project example

action develop(?website)@(start,end) { conditions:

designed(?website)@(t1,t2),implemented(?website)@(t3,t4),tested(?website)@(end,t);

effects:developped(?website)@(end,t);

constraints:designates(?website, WEBSITE),start <= t1 <= t2 <= t3 <= t4 <= end <= t,

end <= start + 18; }

46


State of the Art



action test_in_company(?website)@(start,end){ effects: tested(?website)@(end,t); resources: requires(engineers(),2)@(start,end); constraints: designates(?website, WEBSITE),

end <= t, end = start + 3; }

action test_outsource(?website)@(start,end){ effects: tested(?website)@(end,t); constraints: designates(?website, WEBSITE),

end <= t, end = start + 4; }

47


State of the Art

Problem Modeling : IXTET

IXTET [Ghallab&Laruelle 1994, Laborie&Ghallab 1995] is a temporal and resource planner From IndeXed TimE Table State of the world is represented by a set of valued

attributes (state variables) Relies on time-points as elementary temporal primitives Allow representation of unary, discrete and reservoir

resources

48


State of the Art


State of the world is represented by a set of valued attributes (state variables)

E.g.: color(car):redposition(robot):r1

Conditions are expressed over time intervals

E.g.: hold(position(robot):r1,(t1,t2)) Change is represented via the notion of event

associated with a time-point

E.g.: event(position(robot):(r1,r2), t)

49


State of the Art


Minimal/Maximal delays can be expressed between time-points

E.g.: t in [0, 12]t2-t1 in [3, 5]

Unary and discrete resources can be used over time-intervals

E.g.: use(engineers():2, (t1,t2)) Reservoir can be produced/consumed at time-points

E.g.: consumes(money():100,t)

50


State of the Art


Web-site project example IXTET operators are called task

task develop(?website)(t) { ?website in WEBSITES; timepoint t1, t2; hold(status(?website):designed, (t1,t2)); hold(status(?website):implemented, (t2,t)); hold(status(?website):tested, (t,end)); start <= t1 <= t2 <= t <= end; (end - start) in [0,18]; }

51


State of the Art



task test_in_company(?website)(start, end){ ?website in WEBSITES; event(status(?website):(?, tested), end); uses (engineers():2, (start,end));

(end - start) in [3,3];}

task test_outsource(?website)(start, end){ ?website in WEBSITES; event(status(?website):(?, tested), end); consumes (money():100, start);

(end - start) in [4,4];}

52


State of the Art


Web-site project example The problem is modeled in the same way as a task

resource money() { capacity = 200; }resource engineers { capacity = 5; }

task init()() {timepoint t;explained event(status(website1):(?, init), start);

explained event(status(website2):(?, init), start);explained event(status(website3):(?, init), start);develop(website1)(t);develop(website2)(t);develop(website3)(t);start <= t <= end;t - start in [0,16];

}

53


State of the Art



resource money() { capacity = 200; }resource engineers { capacity = 5; }

task init()() {timepoint t, u;explained event(status(website1):(?, init), start);

explained event(status(website2):(?, init), start);explained event(status(website3):(?, init), start);develop(website1)(t);develop(website2)(t);develop(website3)(t);start <= t <= end;t - start in [0,16];

uses(engineers():1, (u,end)); // One quality engineer onlyu-start in [7,7]; // available first week

}

54


State of the Art

Problem Modeling : O-Plan

O-Plan [Currie&Tate 1991, Drabble&Tate 1994] Open PLANning architecture Hierarchical planning system The language (Task Formalism) allows a rich

representation of time and resources

55


State of the Art


Web-site project example Actions have conditions and effects and can be refined

into sub-actions thanks to schemas

schema develop_website; vars ?website = ?{type websites}; expands {develop ?website}; nodes 1 action {design ?website}, 2 action {implement ?website}, 3 action {test ?website}; orderings 1 -> 2, 2 -> 3; time_windows duration self = 0 .. 18 months;end_schema;

56


State of the Art



schema test_website_in_company; vars ?website = ?{type websites}; expands {test ?website}; conditions only_use_if {free_engineers} = true; effects {free_engineers} = false at begin_of self, {free_engineers} = true at end_of self; time_windows duration self = 3 months;end_schema;

schema test_website_outsource; vars ?website = ?{type websites}; expands {test ?website}; resources consumes {resource money} = 100 kE; time_windows duration self = 4 months;end_schema;

57


State of the Art


Web-site project example The problem is modeled in the same way as a goal

schema

task todo; nodes 1 start,

3 action {develop site1}, 4 action {develop site2}, 5 action {develop site3},

2 finish; orderings 1 -> 3, 1 -> 4, 1 -> 5, 3 -> 2, 4 -> 2, 5 -> 2; resources consumes {resource money} = 0 .. 200 kE overall; time_windows 0 at begin_of 1,

16 at end_of 2; end_task;

58


State of the Art


Time Precedence between actions Minimal/maximal action duration Time windows for events Minimal/maximal delays between events

Schema

dmin dmax

Action

59


State of the Art


O-Plan Resource Typology

Resource

Consumable

Strictly Producible

By-Agent By and OutwithAgent

OutwithAgent

Reusable

Non-Sharable Sharable

Independently Synchronously

Reservoir

Unary

Discrete

60


State of the Art


Resource usage Consumable/Producible resource schema <SCHEMA>; resources consumes {resource <R>}= <q..Q> [at<EVT>]; resources produces {resource <R>}= <q..Q> [at<EVT>];

Reusable resources

schema <SCHEMA>; resources allocates {resource <R>}= <q..Q> [at<EVT>]; resources desallocates {resource <R>}= <q..Q> [at<EVT>];

61


State of the Art

Problem Modeling : ASPEN

ASPEN [Chien&al 2000] Automated Scheduling and Planning ENvironment The main ingredient of the model is the notion of

(hierarchical) activity types. Numerical time State variables Numerical resources (unary, discrete, reservoir).

62


State of the Art


ASPEN Example inspired from [McCarthy&Pollack 2002] State variables: described as a set of possible

transitions

State_variable Position { states = (“patio”,”dining”,…); transitions = (“patio”<->”dining”,

“dining”<->”kit”, “dining”->”living”,

…); default_state = “living”;}

63


State of the Art


ASPEN 3 reasons for an activity to be inserted in the plan

An activity requires a given state for a state variable

Activity

Eat(breakfast)

Reservations:

State Variable: Position must_be kitchen

64


State of the Art



An activity express temporal constraint with another activity of a given type

Activity

Eat(breakfast)

Constraints:ends_before start of Take(medicine) by [0,60]

Activity: Take(medicine)

[0,60]

65


State of the Art



An activity can be decomposed into sub-activities

Activity DailyExercise()

Decompositions:

Exercise1() Exercise2() Exercise3()

66


State of the Art



State_variable status {states = { “init”, “designed”, “implemented”, “tested” };transitions = { “init” -> “designed”,

“designed” -> “implemented”,“implemented” -> “tested” };

default_state = “init”;}

Activity develop {Website w;duration = [0,18];decomposition = { design with (w -> w) a1,

implement with (w -> w) a2,test with (w -> w) a3,where a1 ends_before start of a2,

a2 ends_before start of a3 };}

67


State of the Art



Activity test {Website w;decomposition = { test_in_company with (w -> w) or

test_outsource with (w -> w) };reservations = { status(w) must_be “implemented”,

status(w) change_to “tested” at end; }}

Activity test_in_company {Website w;duration = 3;reservations = { engineers use 2; }

}

68


State of the Art



Activity test {Website w;decomposition = { test_in_company with (w -> w) or

test_outsource with (w -> w) };constraints = { starts_after end of

implement with (w -> w) }}

Activity test_in_company {Website w;duration = 3;reservations = { engineers use 2; }

}

69


State of the Art

Problem Modeling : RAX-PS

RAX-PS [Jonsson&al 2000] Interval based planner No distinction between propositions holding over

intervals and actions taking place over intervals Valued state-variables (same as IXTET or ASPEN) Main ingredient is the notion of token: a given value for

a state variable holding over some time-interval

70


State of the Art


RAX-PS Notion of time-line for a state variable v

Token

tuple < v, P(xP), s, e >

< Attitude, Turn(A,B), 10, 20 >

Engine

Attitude

Idle IdleThrust(B)

Turn(A,B) Turn(B,C)Point(B)

time

71


State of the Art


Each token T is associated a configuration constraint GT called compatibility. This constraint determines which other tokens must be present in a valid plan if the plan contains T and how these tokens are constrained with respect to T.

Usually, GT is a disjunction of basic configurations

72


State of the Art


Example

(Define_Compatibility (ProjectStatus(?w) tested):compatibility_spec

(OR (met-by (ProjectStatus(?w) test_incompany)) (met-by (ProjectStatus(?w) test_outsource)))

(Define_Compatibility (ProjectStatus(?w) test_incompany):constraints

(?duration <- 3):compatibility_spec

(AND (equals (Engineers (+2 Used)))))

(Define_Compatibility (ProjectStatus(?w) test_outsource):constraints

(?duration <- 4))

73


Planners

Planning Techniques:- Planning Graph - SAT Planners and other encodings- Heuristic Planning- Local Search & Repair- Partial Order Causal Link (POCL)- Hierarchical Task Networks (HTN)

Problem Solving

State of the Art

Planning Languages

Basic Tools:- Linear Programming (LP)- Mixed Integer Linear Programming (ILP or MIP)- Satisfiability (SAT)- Constraint Satisfaction Problems (CSP)

74


Problem Solving: Basic Tools

Linear Programming (LP) Input: a set of variables, a system of linear inequations

and a linear objective to minimize

Output: a value for each variable that maximizes the objective function

State of the Art

jijiji

iii

iijji

bxatosubject

xcmaximize

Rxcba

,,

, ,,,

75



Linear Programming (LP) Example

Optimal solution: x=45, y=6.25 (objective = 1.25)

State of the Art

5,45

,21003330,24002450

)50(

yx

yxyx

tosubjectyxmaximize

76



Linear Programming (LP) Efficient algorithms available using the Simplex method Except for some pathological cases, the method

exhibits a polynomial complexity Practical problems involving several hundred thousands

of variables can be solved Many software and libraries available (freeware or

commercial e.g. ILOG CPLEX)

State of the Art

77



Mixed Integer Linear Programming (ILP/MIP) Input: same as LP but some variable are constrained to

be integer

Output: a value for each variable that maximizes the objective function

State of the Art

jijiji

iii

lkijji

bxatosubject

xcmaximize

ZxRxcba

,,

, ,,,,

78



Mixed Integer Linear Programming (ILP/MIP) Example

Optimal solution: x=45.12, y=6 (objective = 1.12)

State of the Art

integerisyyx

yxyx

tosubjectyxmaximize

,5,45

,21003330,24002450

)50(

79



Mixed Integer Linear Programming (ILP/MIP) Problem is NP-hard ... But fairly efficient approaches exist based the

cooperation of LP and a tree search. Basic idea: relax the problem as a LP, solve it, if some

integer variable xl has non integer value vl, branch on xl=[vl] xl=[vl]+1

Survey paper: [Bixby&al 2000]

State of the Art

80



Satisfiability (SAT)

Boolean variables P,Q,R,… true,falseLiteral P,Q,… (un)negated variableClause (PQ),… disjunction of literalsTheory C1C2,… conjunction of clauses

Satisfiability problem: is a given theory valid ?

That is, does it exist an assignment of all the boolean variables in {true, false} such that the theory is true.

MAX-SAT: what is the maximum number of clauses that can be satisfied.

State of the Art

81



Satisfiability (SAT)

Example: (P Q) (Q R) (R P)

A solution: P=Q=true, R=false

(P Q) (Q R) (R P)

State of the Art

82



Satisfiability (SAT) Satisfiability is NP-complete … but efficient methods exist

Complete methods (Davis-Putnam procedure and its improvements)

Approximation methods (randomization and local search)

Can solve problems with thousands of variables Survey paper: [Gent&Walsh 1999]

State of the Art

83



Constraint Satisfaction Problems (CSP)

Problem. Given:

A set of variables X={x1,…,xn}

A finite domain Di for each variable xi

A set of constraints Ci(xi1,…,xik) on some subset of variables described in extension (subset of the cartesian product Di1 …Dik satisfying the constraint) or in intension (e.g. xi=min(xj,xk) )

Find an assignment of all variables in X that satisfies all the constraints

State of the Art

84



Constraint Satisfaction Problems (CSP) Example:

Variables: c {blue, red}

x {1,2,3,4}, y {1,2,3,4}

Constraints: (xy)

(x.y 10)

(c=blue) (x>y) A solution: (x=1),(y=2),(c=red)

State of the Art

85



Constraint Satisfaction Problems (CSP) Problem is NP-complete (SAT can easily be reduced to

a CSP on boolean variables) Efficient resolution techniques using constraint

propagation and tree search Survey papers: [Kumar 1992, Smith 1995]

State of the Art

86


Problem Solving

State of the Art

Planning GraphGraphPlanIPPSTAN

R-IPP

Heuristic PlanningHSPGRTFF

GRT-RMetric-FFSapaTP4

Local Search & Repair

LPGExcaliburASPEN

POCLSNLPUCPOP

parcPLANIxTeTRAX

HTNSipe-2O-PLAN

UMCPSHOP

SAT Planners SAT-PLAN[Rintanen&Jungholt 1999] LPSAT

[Kautz&Walser 1999]

87


Problem Solving: Planning Graph

Planning Graph [Blum&Furst 1997] Planning Graph expansion

State of the Art

0 i-1 i i+1

propositions propositions

actions

88



Planning Graph Mutex relations

State of the Art

InconsistentEffects

Interference Competing Needs

Inconsistent Support

89



Planning Graph Solution extraction

State of the Art

90



Planning Graph Solution extraction

Ad-hoc algorithm (GraphPlan) SAT encoding CSP encoding

State of the Art

91



Planning Graph Some Planning Graph Planners

GraphPlan [Blum&Furst 1997] IPP [Koehler&al 1997] STAN [Long&Fox 1999]

SAT encoding SatPlan [Kautz&Selman 1999]

CSP encoding CPLAN [vanBeek&Chen 1999] GP-CSP [Do&Kambhampati 2000]

State of the Art

92


Problem Solving: R-IPP

R-IPP

IPP extension to resource constrained plans

[Koehler 1998]

GraphPlan based algorithm

The planning graph is annotated with possible interval boundaries for resource level

State of the Art

93



Resource r with possible level in [Qmin(r),Qmax(r)]

State of the Art

0 i-1 i i+1

L(i-1,r)[Lmin(i-1,r), Lmax(i-1,r)] L(i+1,r) ?

94


Problem Solving: R-IPP Resource r with possible level in [Qmin(r),Qmax(r)]

State of the Art

),,1(max riL max(),1(max riL

i-1 i i+1

No action affecting r is applied

95



State of the Art


})(/)({ albetorofamountsettingilevelatactionaal

i-1 i i+1An action a can set theamount of resource to l(a)

L(r):=l(a)

96



State of the Art


})(/)({ albetorofamountsettingilevelatactionaal

)}/)(),1({ max ilevelofproducersofsetexclusivenonSapriLSa

i-1 i i+1

A non-exclusive set ofaction produces r

L(r)+=p(a)

97



Similar calculations for Lmin(i+1,r)

The classical notion of mutex is extended. Two actions are mutex also if: They are of different type w.r.t. a resource (consumer,

provider, producer) They are provider of the same resource Their combined effect is impossible They have contradictory resource requirements

State of the Art

98



Resource boundaries are used to Restrict the number of applicable actions in a given level Allow detection of additional mutexes Prune the search when solving the planning graph

State of the Art

99


Problem Solving

Planning with Resources based on SAT and/or LP/MIP encodings [Rintanen&Jungholt 1999] LPSAT [Wolfmann&Weld 1999] [Kautz&Walser 1999]

State of the Art

100


Problem Solving

[Rintanen&Jungholt 1999] propose an approach similar to R-IPP based on a SAT-encoding Producers and Consumers can be executed in parallel

The search is not restricted to backward-chaining as in the GraphPlan framework.

State of the Art

101


Problem Solving

LPSAT Based on a SAT-encoding of the planning graph

[Ernst&al 1997]

State of the Art

i-1 i i+1

Explanatory Frame Axioms: At(P1,B,i-1) At(P1,B,i+1) SlowFlight(A,B,i) FastFlight(A,B,i)

Actions effects and conditions: SlowFlight(A,B,i) At(Plane,A,i-1) At(P1,A,i-1)

At(Plane,B,i+1) At(P1,B,i+1)

102


Problem Solving

LPSAT SAT extended by LCNF representation and solver:

propositional variables may trigger some linear constraints [Wolfmann&Weld 1999]

(P Q) (Q R) (R P)

P (50x+24y2400)

Q (30x+33y2100)

R (x45)

State of the Art

103


Problem Solving

LPSAT LCNF allows expression of resource requirements

State of the Art

i-1 i i+1

Resource requirements: SlowFlight(A,B,i) (Level(i-1) - Level(i+1)= 15) Refuel(i) (Level(i+1)= 240)

104


Problem Solving

[Kautz&Walser 1999] propose an ILP encoding

Actions a that consumes/produce a resource

One action that reset the resource to its maximal capacity (refuel)

State of the Art

105


Problem Solving

[Kautz&Walser 1999] propose an ILP encoding

State of the Art

ProdaiaiMAX

ConsaiaiMIN

ii

MAXii

ii

iProda

aConsa

iaiii

aqrQ

aqrQ

ks

Qrs

ConsProdaas

aqaqkrr

)0(

)( 1

1

ProdaiaiMAX

ConsaiaiMIN

iii

MAXii

ii

iProda

aConsa

iaiii

aqrQ

aqrQ

MkthatsuchMkMs

Qsr

ConsProdaas

aqaqkrr

,0

0

1

1

1

106


Problem Solving: Heuristic Planning

Heuristic Planning Forward or backward state planning Use of an efficient heuristic h(s)

State of the Art

Cost ofcurrent plan:

g(s)

Initialstate

Goals

Candidateactions

Heuristic cost: h(s’)

s’

107



Heuristic Planning In HSP, the heuristics h(s) is computed by solving a relaxed

problem where all the delete lists of actions are ignored.

s: state, p: proposition, cs(p) estimation of the cost for achieving p from state s

State of the Art

)( pcs0

))((1min )( opPreccspOop

if ps

otherwise{

Goalp

s pcsh )()(HSP heuristics (non-admissible):

108



Some heuristics planners HSP [Bonet&Geffner 1999,2001] GRT [Refanidis&Vlahavas 1999] FF [Hoffman&Nebel 2001]

State of the Art

109



Heuristic Planning with Resources GRT-R [Refanidis&Vlahavas 2000] Metric-FF [Hoffman 2002] Sapa [Do&Kambhampati 2001] TP4 [Haslum&Geffner 2001]

State of the Art

110


Problem Solving: GRT-R

GRT-R: Heuristic planner handling consumable resources in the heuristics

State of the Art


<N=2,R1=5,R2=4>

Initialstate

Goals

Candidateactions

Heuristic cost: h(s’)<N=2, R1=5, R2=1>,

s’

Q1=10, Q2=10

<N=3, R1=2, R2=3>

<N=1, R1=6, R2=2>

111


Problem Solving: GRT-R

The heuristic with resource allows guiding and pruning the search

The heuristics h(s) is computed by regression from the goal state toward state s

Issues: complexity in time and memory for computing the sets of cost vectors

State of the Art

112


Problem Solving: Metric-FF

Metric-FF. Extension to FF [Hoffmann&Nebel 2001] Reminder: FF is a forward search heuristic planner. At

a given state s, the heuristics h(s) is estimated by solving a relaxed problem that ignores the delete lists of operators. This is done in GraphPlan style.

State of the Art


g(s)

Initialstate

Goals

Candidateactions


s’

Solve relaxed problem

113



Metric-FF restricted formalism Conditions: q>C, qC Effects: q+=C (production), q-=C(consumption)

Metric-FF idea: on a resource, something similar to ignoring the delete list is to ignore consumption and only consider production events (monotonicity).

Monotonicity: each proposition true at level i in the planning graph remains true at any level j i

State of the Art

114



As for FF Finding a solution to the relaxed problem is polynomial

(because of the monotonicity) Finding the shortest solution to the relaxed problem is

NP-complete.

A (good) solution to the relaxed problem is computed using similar techniques as in R-IPP

The value of the heuristic is the number of operators in this solution

State of the Art

115



The notion of monotonicity is very generic Extended Metric-FF formalism

Conditions:

linear expressions {<.,=,,>} linear expression Effects:

{:=, +=, -=} linear expression

Transformed into a simpler Linear Normal Form language where a monotonic relaxation is applied

State of the Art

116


Problem Solving: Sapa

Sapa. Heuristic Metric Temporal Planner Focus on durative actions: action A, duration D(A) Notion of time stamped state

State of the Art

tPast Future

pitiPredicatestrue at t

Resourcelevels at t

cj tj

Persistent conditionsto be protected

Scheduled events:- Ends of persistent condition- Change truth value of a predicate- Change a resource

117



Notion of applicable action Result of applying an action to a state

State of the Art

A

tPast Future

pitiPredicatestrue at t

Resourcelevels at t

cj tj

Persistent conditionsto be protected

Scheduled events:- Ends of persistent condition- Change truth value of a predicate- Change a resource

118



Heuristic planning

The heuristics relies on solving a relaxed problem where delete lists and resources are ignored using GraphPlan like techniques

State of the Art


g(s)

Initialstate

Goals

Candidateactions


s’

119



In the relaxed problem Delete lists and Resource usage are ignored

State of the Art

Board(P1,A)

FastFlight(A,B)Deplane(P1,A)

FastFlight(B,C)SlowFlight(A,B)

Deplane(P1,B)Board(P2,B)

SlowFlight(B,C)

Deplane(P1,C)Deplane(P2,C)

ssinit

PartialPlan Relaxed planning graph

Lower bound on the date P1 can be at C(useful to prune the search)

FastFlight(A,B)Board(P2,B)

FastFlight(B,C)Deplane(P1,C)Deplane(P2,C)

Approximation of thedistance from the goal

120



In the relaxed problem, resources are used to adjust the heuristics For a resource R, pre-computation of the action AR that can

increase maximally the level of R (of a level R)

Init(R): initial level of R at state s from which the relaxed plan was computed. Con(R) and Pro(R) are the total consumption and production of R in the solution of the relaxed plan.

If Con(R)>Init(R)+Pro(R), we know that at least

additional actions to produce R will be necessary

State of the Art

R

RProRInitRCon

))()(()(

121


Problem Solving: TP4

Similar approach used in TP4 But the planner is a regression planner, starting from

the goal

State of the Art


Part II: Focus on Constraint-Based Approaches

123



Part II. Focus on Constraint-Based Approaches Reminder: POCL and HTN Planning Constraint-Programming Problem Modeling Problem Solving

Constraint Propagation Search

Overview

124


Problem Solving: Planning Techniques

Partial-Order Causal-Link Planning (POCL) Search in a space of partially-ordered plans Iteratively selects some flaw of the current partial plan

and attempt to fix them until there are no more flaw (solution plan)

Flaws: - unachieved preconditions

- threads on causal links

State of the Art

125



Partial-Order Causal-Link Planning (POCL) Unachieved preconditions

State of the Art

Initialstate

Goal

p

q

q

r

s

u

v

u

v

s

qq

r

p

p

Insert a new action and a causal-link OR

w

126



Partial-Order Causal-Link Planning (POCL) Unachieved preconditions

State of the Art

s

Initialstate

Goal

p

q

q

r

s

u

v

u

v

qq

r

p

Insert a causal-link with an existing action

127



Partial-Order Causal-Link Planning (POCL) Threads on causal-links

State of the Art

Initialstate

Goal

p

q

p

r

s

u

v

u

v

s

pq

r

p

p

Fixes: action promotion OR demotion

128




State of the Art

Initialstate

Goal

p

q

p

r

s

u

v

u

v

s

pq

r

p

p


129




State of the Art

Initialstate

Goal

p

q

p

r

s

u

v

u

v

s

pq

r

p

p


130



Partial-Order Causal-Link Planning (POCL) Some POCL planners

SNLP [McAllester&Rosenblitt 1991] UCPOP [Penberthy&Weld 1992] IXTET [Ghallab&Laruelle 1994]

State of the Art

131



Hierarchical Task Networks (HTN) Hierarchical Operators (see O-PLAN)

State of the Art

Task T(x1,…xn)

precondition: p

q

r

s q

r

s

precondition: v

Method 1 Method 2

132



Hierarchical Task Networks (HTN) HTN Planners work by iteratively

- expanding tasks (choose a method) and - resolving conflictsuntil a conflict-free plan can be found which consists only of primitive tasks

Most of industrial applications of AI Planning use the HTN approach

Provides a strong structuration of the domain to efficiently guide the search

State of the Art

133



Hierarchical Task Networks (HTN) Some Partial-Order HTN Planners

SIPE-2 [Wilkins 1988] O-Plan [Currie&Tate 1991] UMCP [Erol&al 1995]

Often based on constraint propagation [see later] Some Forward-Chaining HTN Planners

SHOP [Nau&al 1999]

State of the Art

134



Partial-order approaches to planning (POCL, HTN) are recognized to be the most flexible for handling time and resources [Smith&al 2000]

Partial order planning is receiving more and more attention [Nguyen&Kambhampati 2001,Geffner 2001]

Constraint-Based Approaches

135



In partial order planning, from the point of view of resources, a partial plan is a temporal network of activities that require/produce/consume some resources.

It’s essentially the same as in constraint-based scheduling


136


Constraint Programming

Constraint-Based Scheduling =

Scheduling + Constraint Programming

Scheduling problems arise in situations where a set of activities has to be processed by a limited number of resources in a limited amount of time


137



Paradigm to solve combinatorial optimization problems

One states the problem in terms of variables and constraints after which, a search procedure is used to find a solution


138



Each variable has an initial domain of possible values ( x [0,…,4] )

Constraints propagate to reduce domains

“x < 3” x [0,…,2]

Each constraint has its own propagation algorithm (reusable)

As propagation isn’t sufficient, a search tree is build


139



Variables, Domains, and Constraints

Problem Definition

Search

Constraint Propagation

initial variables, constraints, and domains

decision making new constraints

backtracking

reduce domains, deduce constraints, contradictions

problem specification

partial solution

Dec. vars: x in [1..3]Dec. vars: x in [1..3] y in [1..3] y in [1..3] z in [1..3] z in [1..3]

Constraints: x - y = 1Constraints: x - y = 1 y < z y < z



x - y = 1x - y = 1

Dec. vars: Dec. vars: x in [2..3]x in [2..3] y in [1..2] y in [1..2] z in [1..3] z in [1..3]




y < zy < z

Dec. vars: x in [2..3]Dec. vars: x in [2..3] y in [1..2] y in [1..2] z in [2..3]z in [2..3]


y < zy < z



y = 2

Dec. vars: Dec. vars: x = 3x = 3 y = 2 y = 2 z = 3 z = 3


y = 2y = 2


140


Constraint-Based Scheduling

Search Scheduling Strategies

Problem Specification / Partial

solution

Constraint Propagation Resource

Demands, Temporal Constraints

Problem Definition Modeling Objects:

Activities, Resources,Temporal Constraints,

Resource Demands


[Baptiste&al 2001]

141


Constraint Programming : Properties


Besides the separation of problem definition, constraint propagation, and search, the Locality Principle [Steele 1980] is important: each constraint must propagate as locally as possible, independent of the existence or non-existence of other constraints.

[Stallman&Sussman 1977] separation between deductive methods (constraint propagation) and search

[Kowalski 1979] distinction between logical representation of constraints and the control of their use: Algorithm = Logic + Control

142




Global Constraints One of the strong points of CP tools are the so called

global constraints (Opposed to constraints like x < y.) Example is the “all-different” constraint where you have

n variables which all should take a different value:

)(,/],...,[, ji xxjinji 1

143



O(n2) binary constraintsWeak constraint propagation

One global constraint(Easier modeling)

Strong constraint propagation[Regin 1994]

V12

V6

V3V9

V2

V4

V10

V8

V11

V5

V1

V7


144



Constraint Programming : Search

Search Heuristics Variable and value selection heuristics. Choose variable to assign a value to and then choose a

value for this variable (instantiate the variable). Typical strategy

Choose most constrained variable, i.e., a variable that is difficult to instantiate; start with the difficult parts (this before they get even more difficult).

Choose least constraining value, i.e., a value that leaves as many values as possible for the remaining uninstantiated variables.

145




Variable selection heuristics choose variable with smallest remaining domain choose variable with maximal degree in constraint graph

(most constraints defined on it)

146




Value selection heuristics choose value that participates in highest estimated

number of solutions. Number of solutions are estimated by counting the solutions to a tree-like relaxation or the original constraint graph.

choose minimal value

Recall: In general decisions correspond to additional constraints.

147




Search Tree Exploration Depth First Search (DFS). The traditional search

strategy of CP. Best First Search (BFS). Choose the node with the best

minimum cost. Limited Discrepancy Search (LDS). Divides the search

tree into strips, trying to stick close to the heuristic. … and several others like Depth-Bounded Discrepancy

Search, Interleaved Depth-First Search.

148




Optimization In case an objective function is present, one tries to find

the best solution. Such optimization can be done in several ways When a solution with cost c is found, restart search to find

a solution with cost c-1. Do dichotomization. Besides upper bounds coming from

solutions, when being complete one can also calculate lower bounds.

When a solution with cost c is found, stay in this solution node, add a “cut” that says the cost needs to be at most c-1, and continue search.

149


Problem Modeling

Scheduling is the “problem of allocating scarce resources to activities over time” [Baker 1974]

Typology of Activities Temporal constraints Resources


150


Problem Modeling : Activities

Non-preemptive activities


Activity A

Start time: s(A) End time: e(A)

Processing time: p(A) == e(A)-s(A)

t

1 Activity 3 numerical Variables

151



Notations on current domain of activity time window


Activity A

t

smin(A) emin(A)

emax(A)smax(A)

152



Preemptive activities


Activity A

Start time: s(A) End time: e(A)

Processing time: p(A) = pi e(A)-s(A)

tp1 pi...

153



Preemptive activities Preemption at fixed dates (break calendar)


Processing time: p(A) = pi = e(A)-s(A) - breaks(A)

Activity A

tp1 pi...

M T W T F S S M T W T F S S M T W T F S S M T W T F S S M T W T

1 Activity 3 numerical Variables

154



Preemptive activities Preemption by other activities


Activity A

tp1 pi...

Activity B Activity C

1 Activity 1 Variable Set of dates

155


Problem Modeling: Temporal constraints

Generally expressed as ti - tj [dmin, dmax] where ti, tj : start or end time point of some activities

dmin, dmax: minimal and maximal delay

Generic enough to model release dates: start(A) [r,+) deadlines: end(A) (-,d] precedence: start(B)-end(A) [0,+)


156



Simple Temporal Problem (STP)

[Dechter&al 1991] Distance Graph representation


ti

tj

-dmin

dmax

157



Arc consistency ( Single Source Shortest Paths) ensures a backtrack-free search n=#time-points, p=#temporal constraints Memory: O(p) Time: O(n.p) (with negative cycle detection)

Path consistency ( All Pairs Shortest Paths) Memory: O(n2) Time: O(n2) Provides all pairs distances (useful for heuristics)


158



Problem Modeling : Resource Usage

Resource quantities q(U) are decision variables The same activity may use several resources

A

U1: A requires Machine1

U2: A requires(q2) Workers

U3: A consumes(q3) Memory

159



Problem Modeling : Resource Usage

Resource usage follow the “constraint protocol” In particular, they can be used in meta-constraints

[A requires Machine1]

[A consumes(q2) Memory] [B requires(q3) Workers]

160



Problem Modeling : Decision Variables

Start, end and processing time of activities:

s(A), e(A),p(A)

Required quantities of resources:

q(U)

From the standpoint of constraint-programming, the limited capacity of resources imposes some global constraint on the decision variables of the problem

161



Problem Modeling : Objective functions

Often the objective is to minimize a function

f(e(A1),…,e(An))

Makespan: f = max(e(Ai)) Total weighted flow time: f = wi.e(Ai) Maximum tardiness: f = max(Ti); Ti=max(0,e(Ai)-i) Total weighted tardiness: f = wi.Ti

Total weighted number of late jobs: f = wi.Ui

Earliness/Tardiness costs

162



Problem Modeling : Objective functions

Other common objective functions

Minimum transition costs Minimum peak resource usage Schedule maximum weighted # of activities Schedule on a minimum weighted # of resources

163



Problem Modeling : Scheduling Problem

Given: A set of resources A set of activities A set of temporal constraints on/between activities A set of resource usage

An objective function f Find an instantiation of the decision variables that

satisfies all the constraints and minimizes f

164



Problem Solving

A.I. Planning and Scheduling techniques start to converge into a “POPS” (Partial Order Planning and Scheduling) concept Partial Order Planning is “Reviving” “Partial Order Scheduling” is most of what Constraint-

Based Scheduling is about

165



Problem Solving

POP and POS work on the same type of structure: a temporal network of operators/activities POP focus on operator/activity generation constraints POS focus on time and resource constraints

The fundamental idea is that in both case, a solution plan or a solution schedule can be characterized by a set of local properties

These local properties can be enforced by constraint propagation and/or by branching in the search tree

166



Problem Solving

Example of such a local property in POP: The Necessary Truth Criterion [Chapman,87]

Pre: free(A) ?

Establisher: +free(A)

Clobberer: -free(A)

White knight: +free(A)

167



Problem Solving

In what follow, we identify the local properties that are used in Constraint-Based Scheduling to characterize solution schedules and describe

1. the propagation and

2. the branching schemes

based on these local properties

168


Problem Solving : Global Constraints

2)1( nn

)()'(

or),'()(

AsAe

AsAe


What’s happening inside ? Key principle is that a resource knows the set of

activities that requires it (start, end, processing time)

For instance a unary resource takes care of thedisjunctive constraints of the form

169




What’s happening inside ? When adding an activity requiring a resource, the

resource automatically takes care of the relations with all other activities requiring the same resource

Remark: One can add activities during search

Note however that sometimes you need to know that all the activities that will require the resource are known to do any propagation

170




What’s happening inside ? We have several types of constraints that all make sure

the capacity of the resource is not exceeded at any point. These constraints differ in the strength of the propagation they induce: timetables precedence energy

disjunctive constraint balance constraint

edge-finding

In general: more propagation = more effort

These constraints are global constraints, they “globally” consider the set of activities on a resource

171



Different algorithms available to propagate resource capacity (timetable, disjunctive, edge-finder, balance, etc.)

Several algorithms may be used in cooperation on the same resource

A resource knows the set of activities that uses it As soon as something changes on the resource

(change of activity start/end/processing time, change of resource demand, new resource usage added), propagation algorithms are triggered


172



A resource is said to be closed if no additional resource usage will be inserted into the current schedule lower in the search tree

When a resource is closed, stronger propagation can be inferred


LINIT=0 +q -q

173



In mixed A.I. Planning & Scheduling, a resource can be considered closed if for example : No operator use this resource; or In hierarchical planning, it corresponds to an already

processed abstraction level; or The decision has been taken to close the resource on

the current sub-tree :


close R insert an operator using R

Forbid insertion of any operator using R. Stronger pruning.

174


Algorithms based on time windows


Time-tabling [LePape 1994] Interesting activities A are those for which

emin(A) > smax(A).

Between smax(A) and emin(A) we know A will be executed.

mins

maxe

mine

maxs

175


qmins

maxe

mine

maxs

Q

t

Aggregated necessary demand

QMAX



176


mins

maxe

mine

maxs

Q

t

Aggregated necessary demand

QMAX



minemins

177




Time-tabling

C() E()

Time Bounds Level Time Bounds Precedence Energy Precedence

Not First/Last

178




Time-tabling Used for large problems

Worst case O(n) algorithm

Incremental

Smart about when to propagate and when not, group propagation, etc.

In practice a very fast algorithm

179




Disjunctive constraint Considers pairs (A, B) of activities If two activities can not overlap because of the capacity

they require, and one, say A, can not be scheduled before the other, then New time bounds are deduced A new precedence constraint is deduced

180


)()( minmax BeAs

mins

maxe

mine

maxsA

mins

maxe

mine

maxsB

)()(

)()(

minmin

maxmax

BsAe

BsAe

mins

maxe

mine

maxsA

mins

maxe

mine

maxsB



Disjunctive constraint on unary resource [Erschler 1976]

181




C() E()


Not First/Last

Disjunctive constraint

182




Disjunctive constraint Works on unary, discrete, and state resources More propagation than timetable constraint on unary

resources Low memory cost Needs fake activities to cope with resources not

available at certain times Algorithm : O(n2)

183


][)]()()(}){([,, minmax AAppsAeA



Edge-Finding [Carlier&Pinson 1990] [Nuijten 1994] [Baptiste&LePape 1995]

Detecting First Among n Activities on Unary Resource

184




Edge-Finding

B

C

A

6 16

7 15

4 16

p=4

p=5

p=2

B

C

A

6 16

7 15

4 7

p=4

p=5

185




Similar reasoning for detecting Last, Not First, Not Last with respect to a set

Edge-Finding can be extended to discrete resources [Nuijten 1994, Baptiste&LePape 1996]

C() E()


Not First/Last

186


])()()())()(.([},{

))max(),min([

maxmin BAC

AeBs

MAX CWBWAWAeBsQ

)]()([ BeAs



Energetic Reasoning [Erschler&al 1991] Discrete resource: {A,B} Acts,

187


C1

C2

A

6 16

5 15

16

p=1

p=3

p=4

B16

p=4

4

6

1

2

4

4

C1

C2

A

6 16

5 15

12

p=1

p=3

p=4

B16

p=4

4

8



Energetic Reasoning

188




Energetic Reasoning

C() E()


Not First/Last

189




More Propagation based on time windows Preemptive resource constraints: mixed edge-finding

[LePape&Baptiste 1998] Activity cannot be nth [Nuijten&LePape 1998] Energetic reasoning [Baptiste&al 1999] Multi-machine propagation [Sourd&Nuijten 2000]

190



Algorithms based on relative positions

Algorithms based on analyzing time bounds of activities propagate few (or even nothing) when the problem is not tight

Example: On discrete resources, for the time-table constraint, if for all activity of the schedule: emax - smin 2.pmin, the time-table constraint propagates nothing

t

smin emin

emaxsmax

191




These propagation algorithms do not directly take into account precedence relations (but only through their effect on time bounds)

In PO Planning

Time bounds of operators are loose

Extensive usage of precedence relations between operators in the partial plan

Traditional resource propagation algorithms do not prune the search space enough

192




Example: Blocks world. Limited number of hands (2) modeled as a reservoir of capacity 2 and initial level 2

-1-1

-1

+1

+1

+1

193




Analyze precedence relations on the left side of the propagation rule (C())

Based on a Precedence Graph that collects and maintains precedence relations between activities:

Initial statement of the problem Precedence constraints inside a planning operator Search decisions (e.g.: causal link, threat, ordering

decision on resources) Discovered by propagation algorithms (disjunctive

constraint, edge-finding, energetic reasoning, emaxsmin).

194




Precedence Graph

Events x=(tx,qx) where the availability of a resource changes

Arcs : Two kind of arcs: tx<ty and tx ty

Transitive closure is incrementally maintained

+2+2

-1[-10,-5]

-2 +2

Activity

EventProductionConsumption{

{Arc <

195




Partition for an event x in a precedence graph

E(x)

x

U(x)

BE(x)

BS(x)

AE(x)

AS(x)

B(x) A(x)

196




Energy Precedence Propagation [Laborie 2001]

Does not assume the resource to be closed

For any solution:

In a partial schedule, we can decline this property to perform constraint propagation

YMAXY

QYpYqYsXs

XBUX

/))().(())((min)(

),(,

197




Energy Precedence Propagation [Unary,Discrete]Discrete resource: X U,

Y

MAXYQYpYqYsXsXB ]/))().(())((min)([)]([ minminminmin

B

X

0

p=4, q=2

C p=4, q=1

p=2q=1

E

p=2q=2

QMAX=2

D

B

X

0

p=4, q=2

C p=4, q=1

D

p=2q=1

E

p=2q=2

9

198




Energy Precedence Propagation

C() E()


Not First/Last

Worst-case complexity: O(n2)

199




Balance Constraint [Cesta&Stella 1997,Laborie 2001] Assumption: when applied to a reservoir, all producers of

the reservoir are known in the current schedule Local property: for a given event x in a solution schedule,

we can compute the level of the reservoir just before x at date t(x)- as:

A schedule is a solution on the reservoir iff

In a partial schedule, we can decline this property to perform constraint propagation

)(

)()(xBSy

INIT yqLx

MAXQxUx )(, 0

200




Reservoir Balance Propagation Given an event x, using the graph, we can compute

an upper bound on the reservoir level at date t(x)- assuming:All the production events y that may be executed strictly

before x are executed strictly before x and produce their maximal quantity q(y);

All the consumption events y that need to be executed strictly before x are executed strictly before x and consume their minimal quantity q(y);

All the consumption events that may be executed simultaneously or after x are executed simultaneously or after x

)(x

201


INITLx)(

x

U(x)

E(x)BS(x) AS(x)

BE(x) AE(x)

)()(

max )(xUxBy

yp

)(min )(

xBSyyc


202




Reservoir Balance Propagation Similar bounds can be computed:

Lower bound on the reservoir level at date t(x)- Lower/Upper bound on the reservoir level at date t(x)+

For symmetry reason, we focus on Given , the reservoir balance algorithm discovers:

FailuresNew bounds for required quantities of resources q(x)New bounds for time variables t(x)New precedence relations

)(x

)(x

203




Reservoir Balance Propagation Discovering failures: As soon as: , no

solution exist that satisfy the precedence relations Property: the rule is sufficient to

ensure the soundness of the search on a reservoir w.r.t. the reservoir underflow.

A symmetrical property exists for reservoir overflow based on .

][])(,[ failxx 0

0 )(, xx

)(x

204


E(x)

U(x)

BE(x)BS(x)

[-1,-1] [+1,+1]

[+0,+10] [-5,-5]

[-5,-4]

[-5,-5]

[+1,+1]

[+1,+3]

[+1,+2] [-2,-1]

LINIT=0

-(x)= 0 + (2+1+10+1) - (1+5+5+4) = -1 < 0 !!!


205




Reservoir Balance Propagation Safe event: An event x is said to be safe if and only if

Property: If all the events are safe, then any instantiation of the time variable that satisfies the precedence constraints also satisfies the reservoir constraint. In other words, the current partial order is safe and the reservoir is “solved”.

00

)(,)(

)(,)(

xx

QxQx MAXMAX

206




Reservoir Balance Propagation Reservoir balance equation:

In case the first part of the equation is such that:

It means that some events in will have to be executed strictly before x in order to produce at least:

)()(

max)(max )()()(

xUxBEyxBSyINIT ypyqLx

0 )(

max )(xBSy

INIT yqL

)()( xUxBE

)(max )()(

xBSyINIT yqLx

207




Reservoir Balance PropagationSome events will have to be

executed strictly before x in order to produce at least

Let the set of production events in

sorted by increasing minimal time Let k be the smallest index such that: Then:

)()( xUxBEz

)(x

),...,,...,( ni zzz1

)()( xUxBE

)()(max xzpk

ii

1

)()( xtzt k

208


E(x)

U(x)

BE(x)BS(x)

[-2,-2] [+2,+2]

[+0,+10] [-5,-5]

[-5,-4]

[-5,-5]

[+1,+2]

[+1,+3]

[+1,+2] [-2,-1]

LINIT=0

(x)= 0 - (2+10) + (2+5+5+4) = 4

[+1,+2]


209




Reservoir Balance Propagation In AI Planning applications, if some producers of the

reservoir may still be inserted in the partial plan: transform the balance algorithm into a real truth criterion:Some events will have to be executed

strictly before x in order to produce at least

OR

Some operator containing a producing event z will have to be inserted such that t(z) < t(x)

Such a “truth criterion” is described in [Laborie 2003]

)()( xUxBEz )(x

210




Reservoir Balance Propagation

C() E()


Not First/Last

Worst-case complexity: O(n2) O(n3)

211


Unary

Discrete

Reservoir

Time-tabling Energe-

ticReason.

Edge-Finding

O(n)

O(n log n)

O(n3)O(n2)

Level-Based

Energy-Based

Time-Bounds analysis

Preced.Energy

Balance

O(n2)

O(n2)O(n3)

Precedence analysis

Disjunc-tive


O(n2)

O(n log n)O(n2)

212


Search Techniques

Branching Schemes Search Heuristics Beyond the Basic Search Scheme


213



Branching Schemes

Example of Branching Schemes :

Event Pair Ordering Setting Times

214



Branching Schemes

Property : In a solution for a unary resource, all the pairs of resource usage are ordered.

Activity Pair Ordering (Unary Resource) : Select a not-ordered pair of resource usages : (u,v) U Branch on : [e(u) s(v)] [e(v) s(u)] Needs a precedence graph to maintain the sets of not-

ordered pairs of resource usage

215



Branching Schemes

Property : On a capacity resource, a partial schedule where all the events are safe is a solution.

Event Pair Ordering (Discrete, Reservoir, etc.) : Select a critical pair of events (x,y) on a resource R

(event = start or end time-point of an activity using R) Critical Pair of events : not-ordered pair of event : (x,y)

such that neither x nor y is safe (in the sense of the balance constraint)

Branch on : [t(x) < t(y)] [t(x) t(y)]

216



Branching Schemes

Property: in a solution on a discrete resource, there is no Minimal Critical Sets.

Minimal Critical Set Ordering [Laborie&Ghallab 1995, Cesta&Oddi 2002] Select an MCS on a resource :

Branch on the resolvers of this MCS :

[t(e1) t(s3)] [t(e3) t(s1)] [t(e3) t(s2)]

s1

s2

s3

e1

e2

e3

q1+q2+q3 > Qq1+q2 Qq1+q3 Qq2+q3 Q

217



Branching Schemes

All previous branching schemes are complete Setting Times (All resources) :

Select an activity a in the schedule Branch on : [s(a) = smin(a)] [s(a) > smin(a)] Dominance rule relying on constraint propagation :

instead of posting [s(a) > smin(a)], activity a is marked as non-selectable until its start time has been updated by propagation.

218



Search Heuristics

Slack-Based Heuristics [Smith&Cheng 1993] Choose a resource where the slack of any subset of

potentially conflicting activities on that resource is minimal. Slack of a subset S is maxA S (emax(A)) – minA S (smin(A)) – sumA S p(A)

Sequence that resource / Take a sequencing decision on that resource.

Choose pair of unsequenced activities with minimal maximal slack. Maximal slack of A and B is max(emax(A) – smin(B), emax(B) – smin(A)) – p(A) – p(B).

Choose sequencing that leaves maximal slack.

219



Search Heuristics

Texture-based Heuristics [Beck&al 1997] Texture measurement: dynamic analysis of a search

state to reveal problem structure Heuristic: based on the revealed structure, find “most

critical” part of the search state and take a decision to reduce criticality

The texture measurement and the heuristic to take decisions are separate.

Note: for efficiency reasons one might want to make more than one decision before recomputing texture measurements

220


Search Heuristics

Example of Texture-based Heuristics: SumHeight Heuristic Texture measurement: a probabilistic estimate of the

extent to which activities are competing for each resource and time point (“contention”)

Find the most critical (highest contention) resource and time point

Take a decision to reduce contention


221


Search Heuristics

time

individualdemand

for R

smin emin smax emax


The SumHeight Texture Measurement: Individual Demand

222


Search Heuristics

time

aggregatedemandfor R2 R2

R2

R2


223



Search Heuristics

The SumHeight Decision Find the resource, R*, and time point, t*, with highest

contention Find the 2 activities not sequenced with each other with

the highest individual demand for R* at t* Post a precedence constraint between them Complexity:

R = # resources, n = # activities per resource O(R.n.log(n)) for texture calculation O(n2) for finding decision after texture calculation

224


Beyond the Basic Search Scheme

Shaving [Torres&Lopez 2000] Perform some kind or forward checking that exploit the

interactions between resources Use some global propagation algorithms to determine if

there exist a schedule that verify a property p E.g. Try posting A B on a unary resource ...

If no solution schedule verify p then p must be verified in any solution. Thus it can be inferred. … if global propagation algorithms fail, then infer the opposite

constraint B A


225



Shaving. Example of application: Time-Bound Shaving Activity A, Select such that emin(A) emax(A)

If propagating e(A) < leads to a dead-end: infer e(A) If propagating e(A) > leads to a dead-end: infer e(A)

Perform a dichotomic search on [emin(A), emax(A)] to compute the best adjustments

Other shavings: Precedence, Ranked First/Last activities


226



Shaving. Example (Tested on ILOG Scheduler): Famous Job-Shop problem mt10 (1010) Strong constraint propagation (disjunctive, edge-finder,

precedence energy) Strong Shaving (time-bounds, precedence, rank

first/last). Shave until a fix point is reached. Optimality proof requires:

2 choice points !!!! About 2mn CPU time


227



Probing Main idea: At each node of the search, solve a relaxed

version of the current schedule and use this solution to guide the search Relaxed problem: for example only keep temporal constraints

[ElSakkout&Wallace 2000] or cost relevant activities [Beck&Refalo 2001]

If the solution to the relaxed problem cannot be extended to a solution, identify the discrepancies and branch on the different alternatives to solve these discrepancies.

If the relaxed problem can be solved to optimality, it provides a lower bound on the objective function


228



Probing [ElSakkout&Wallace 2000, Beck&Refalo 2001] Example: Job-shop with Earliness/Tardiness Costs of

last activity


229



Probing During the search, maintain at each node the optimal

solution of a linear relaxation of the problem (no resource constraints)


Cost:

Precedence:

Time windows:

imimiii ddptxtcy ,,

mimiiii ptxddecy .,

jijiji asxas ,max,,min

1 jijiji xptx ,,,

Minimize subject to:i

iy

230



Using the relaxed optimal start time (from LP), build textures

Identify a peak (where the texture > capacity) Select a pair of activities, A and B, at the peak Branch: (A B) (B A) New precedence constraints (as well as precedence

discovered by constraint propagation) are used to update the LP relaxation


231



Properties that contributed to the industrial success of Constraint-Based Scheduling The paradigm clearly separates the constraints

(semantics, propagation algorithms) from the search space exploration (branching schemes, heuristics)

Constraint propagation allows dramatically pruning the search before everything gets instantiated

Each constraint is an autonomous algorithm, the paradigm is extensible: new application-dependent constraints can easily be implemented

Conclusion

232



Properties that contributed to the industrial success of Constraint-Based Scheduling Constraint propagation algorithms often rely on some

bounds that can be seen as the optimal solution to a relaxed (sub)problem

In optimization problems, the search tree is pruned by computing some lower bound on the objective function, usually it corresponds to the optimal objective function of a relaxed problem

Powerful heuristics also often rely on analyzing a good solution to a relaxed problem

Conclusion

233



Conclusion

Propagation Search

Real problem

Relaxed (sub)problems

Optimal solution

Good solution

bounds bounds heuristics

234


Constraint-Based AI Planning

Conclusion

Propagation Search

Real problem

Relaxed (sub)problems

Optimal solution

Good solution

bounds bounds heuristics

Part II

Part I

Planning with Resources

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