Scheduling and Planning With Constraint Logic Programming

7/21/2019 Scheduling and Planning With Constraint Logic Programming

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COSYTEC 1Copyr ight 1995, COSYTEC SA

Constraint Logic Programming

COSYTEC SA


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Overview

The problem what do we discuss

how do we express a problem

Applications

Coffee

Break

examples of the use of the approach

To probe deeper


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Note

Tutorial with same name at PAP94 Structure is kept

Methods have changed

New constraints

cycle

more abstraction, more propagation

machine choice

setup time problems

More applications


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What we will discuss

Planning, scheduling, assignment Emphasis on detailed scheduling

Emphasis on constraint

Modelling techniques se o cons ra n s

Logic Programming Used as basis

Alone not sufficient


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What we will not discuss

System Considerations Object/Database Design

Graphical Interface Design

Quality Assurance

Methods used inside constraint solvers

Classical TechniquesAlgorithms

Strategies


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Part 1: The problem

Terminology where we define the different terms and notations

where we describe different classes of problems

Classical results where we present different results from Operational

Research (OR) on scheduling and planning


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Terminology

What do we mean by Scheduling

Assignment

The main concepts esources

Tasks

Jobs

me Cost


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Definitions

(Production) Planning what to produce in which quantity in which period

period in which quantity

driven by demand or forecasts

when to produce

detailed plan when to start production, when to stop

(Resource) Assignment how to produce

which person (machine) will do what at which time

often on ad hoc basis


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Resources

Renewable Typical examples:

manpower

Used during t ime period

Consumable Typical examples:

raw ma er a money

utilities

se at a certa n t me po nt

Constraints on overall use in a time period


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Resources (2)Renewable Resource

tonsuma e esource

t


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Tasks

Elementary operation during time period May or may not use resources

u resource use

partial resource use

Tasks may be split (interrupted) preemptive scheduling


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Jobs

Set of tasks belonging together Sequence of tasks may be pre-determined

prece ence cons ra n s e ween as s

Jobs often relate to manufacturing of a product or

an order


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Time

Discrete/continuous

Granularity

ann ng or zon

Problem size: number of time points

1 day 1 week 1 month 1 year

second 86400 604000 - -

minute 1440 10800 44640 525600

shift 3 21 93 1095


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Time and tasks

Start Date t i the actual starting t ime of a task

i the actual end time of a task

Duration p i Ci-t i

may vary depending

on the resource used

on the start date

etc


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Due/release date

Due Date d i the time when the task should be finished

i the timepoint after which the task can be started

These dates may be hard constraints (they must be respected)

soft constraints (they may be violated at a cost)


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Earlyness/lateness

Earlyness the amount of time a task is finished before its due date

-,

Lateness Ti the amount of time a task is finished after its due date max , i- i

Latency q i the period between the end of the task and the end of the

schedule max(0,Cmax-Ci)

Setup the time required between two consecutive tasks on the

same machine


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Time points and periods

Graphical view as seen in Gantt charts

earlyness

duration

Resourcesetup

lateness

start date end date

T1 T2

due daterelease date

time

due dateT1,T2 T2 T1


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Cost

Overall Enddate Cmax project duration

Cmax = max{Ci}

Maximum Lateness Tmax e max mum e ay o a as

Tmax = max{Ti}

Sum of Lateness

a weighted sum of all delays Sum w iTi


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Cost (2)

Earlyness indicates stocks of finished products

0/1 variable indicating delay

important for jit (100% customer satisfaction)

Resource Utilization important when resource choice is possible

cost of changing/cleaning machines

Machine Downtime t mes w en a resource s e

Stock Levels


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Classification

Comes from OR methodology Classification of pure problems

Often encountered in reality whenconsidering only main constraints Project Planning

Machine Shop

Flow Shop

Job Shop

Open Shop


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Project planning

Precedence or sequence constraints No resources

Cost: optimize Cmax

Very large problem instances (> 10000 tasks)

,

Deterministic algorithms


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Machine shop

Characteristics Different resources

Each tasks requires one resource during its duration

No choice of resources

No preemption

Subclasses:

ow op Job Shop

Open Shop


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Notation

Shorthand problem notation [n/m/F/D] n jobs

F; J Type e. g. Flow shop/ Job Shop

D optimization criteria, e. g. Cmax


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Flow shop

Characteristics: Precedence constraints between tasks of a job

sequence

duration of tasks on resources may vary

[n/2/F/Cmax]

Johnsons algorithm

max

polynomial under dominance criteria


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Job shop

Characteristics: Precedence constraints between tasks of one job

Deterministic subclasses

[n/2/J/Cmax] ac son s ru e

Job-shop with 2 jobs


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Open shop

Characteristics: No precedence constraint between tasks

Deterministic Subclasses

[n/2/O/Cmax] onza ez a n a gor m

Many standard methods do not work due tolack of precedence constraints

complexity of choice functions


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Complexity results

Most machine shop problems are NP-hard difficult problem instances exist

still, good solutions can be found

application specific knowledge is useful

basis for strategies for more complex problems

Complexity independent of cost function

research centered on Cmax cost


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OR techniques

Linear/Integer Programming

Heuristic Algorithms

Decomposition methods Hierarchical

Structural

Temporal/spatial

Branch and bound

Simulated annealing

Tabu search

Relaxation methods


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Linear/Integer Programming

Expression of scheduling problems with linear programming

Often poor result due to non-convexity of

solution space a n a cos

deep backtracking

poor cuts

Seldom used in practice for detailedscheduling

planning


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Heuristic Algorithms

Progressive building solutions by addingtasks/jobs one at a time

Good solutions for weakly constrained

problems Bad results for strongly constrained

problems

Heuristics should take constraints intoaccount ynam c, not stat c or er ng


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Decomposition techniques

Cut problem into more manageable parts helps handle large/complex problems

projects and sub-projects

bottom-up and top-down requ res cer a n pro em s ruc ure

Structural considering different degrees of freedom independently

example: separating scheduling/assignment

Temporal/Spatial

number of resources


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Branch and bound

Create successive sub problems byenumeration on variables

pruning of branches

lower bound approximation Standard OR technique

Search strategies must be defined carefully

High development effort


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Neighborhood search

Search by finding initial solution and improving it

modification function

cost evaluation

steepest ascent

hill climbing

s mu a e annea ng

tabu search

genetic algori thms

Local optimization


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Neighborhood search (2)

Constraint handling Constraints expressed in cost

Good for additive costs

Local changes which improve costs Difficult for very constrained problems

Finding ini tial solution

Admissible modifications


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Relaxation methods

Solving simpler problem helps findingsolution to complex problems

Obtain lower/upper bounds

Initial solutions


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Importance to CLP

Defines areas in which CLP approach should(not) be used

constraint programming

Application use of techniques Reference results


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Part II: Methodology

Constraints: the tools

Solution approach

Strategies


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Constraints and how they work

Finite Domain Solver based on local consistency techniques

incremental

requires enumeration

based on Simplex method

complete solver

so ve orm

Specialized Solvers (not discussed) Boolean Solver

List Constraints

Set Constraints

COSYTEC


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Finite domains

Variables which range over sets of admissible values (integers)

which link variables and restrict the values which can beassigned to them

earc ra egy which determine

the order in which variables are assigned

the values which they take

COSYTEC


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Finite domain basics

Domain definition

Numerical constraints nequa y cons ra n s

Global constraints

Cumulative Diffn

Cycle

COSYTEC


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Domain definition

A variable or list of variables are defined

Different types of domains exist

Examples X :: 0..100,

[X1, X2, X3] :: [1,2,4,6,7,11]

Operations on domains enumerate domain values

get information about domains

COSYTEC


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Inequality constraints

Link two variables (simple case) example: X #>= Y+10

consistency technique partial lookahead

constraint is woken whenever one of the domains change

constraints) example: X+Y+3*Z #>= T+5*U

propaga on v a oma n c anges

generalization of simple case

COSYTEC


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Global Constraints

very high level building blocks

mix and match combination

not specialized constraints, but generalconstraints

COSYTEC


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Cumulative constraint

Graphical Introduction cumulative([O1,O2,...],[D1,D2,...],[H1,H2,...], L)

LHi

i

Oi

COSYTEC


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Cumulative (2)

Mathematical Form min = min{Oi}

= + -

i [min, max]

Hj

L for all j with Oj

i Oj

+Dj

-1

Global constraint handling ~ 20 different incremental algori thms

20000 lines in C

Only most simple form shown here

COSYTEC


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Diffn constraint

Constraint for n-dimensional rectangle

overlap - -

space and f it in a n-dimensional area

Form 11, 12, 13,..., 11, 12, 13,.. ,... , 1, 2,...

Origin Oij

Length L ij

j ij ij

Useful for 2D placement/ packing

3D placement/ packing

Vehicle loading

COSYTEC


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Two dimensional placement

Domain

Limit

Obj2

E2

Obj1L12

Obj3O12O32 L11

E1O11Limit

COSYTEC 48C i ht 1995 COSYTEC SA


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Cycle constraint

Find cycles in directed graphs

Example: 8 nodes, 3 cycles

1 3 6

2 45

7

8

cycle(3,[3,2,6,1,4,5,8,7],...) : cycles written as permutations

COSYTEC 49Cop r ight 1995 COSYTEC SA


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Tour generation

Nodes are cities and resources

One resource per cycle

All cities in one cycle are visited by oneresource

C6

C1C8

C7R1

C4C2 C8 R2

R3C3

COSYTEC 50Copyr ight 1995 COSYTEC SA


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50Copyr ight 1995, COSYTEC SA

Cycle constraint in scheduling

Nodes = tasks and resources

Domain variables = successor nodes = next

Domains = possible successors

= Weight of nodes = work time of resource

Special assignment nodes = resources

Origin of nodes = start times



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Solution approach

Expressing scheduling problems with CLP

Use of the finite domain solver con nuous so ver use or p ann ng pro ems

Problem statement defines

Variables Constraints

Strategy

expressive power of constraints search concept part of the language



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Variables

Decision variables variables correspond to decisions in the schedule

defines size of domains

large domain = large search space

Calendars include/exclude particular dates

use of explicit domains

mapping real dates on schedule dates Release/ Due dates



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Copyr ight 1995, COSYTEC SA

Example

Link between calendar and domains

Calendar

Monday

0 2 4 6 8 10 12 14 16 18 20 22 0 2 4 6 8 10 12 14 16 18 20 22

Tuesday

Domain (explicit domain)

Domain (mapping)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30



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py g ,

Constraint types in scheduling

Precedence

Sequence

Disjunctive Machines

Cumulative Machines

Machine Choice

Preemption

Cost



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Methodology

How to express different constraints

Alternative models

Use of global constraints

Building blocks for applications



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Precedence

One task must be finished before another

Expressed as inequalities

Constraints between variables denoting start dates or

between variables for start and end dates

task a task b

Da

Sb #>= Sa + Da

a a b

Sb #>= Ea



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Sequence

Sequence constraints l ink start or end dates Start-Start

-

End-Start

End-End

Efficient handling by constraint propagation but problems with very large sets of constraints



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examples

Example problems simple

not realistic

CHIP Modelling ac ua co e

simple structure

Results

to help understanding

Example



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Project planning

Simple project example

B E

A G

3

4 2

2

C D F

3 2 4



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Modelling

[Sa,Sb,Sc,Sd,Se,Sf,Sg] :: 0..100,

[Da,Db,Dc,Dd,De,Df,Dg] = [3,4,3,2,2,4,2],Sb #>= Sa + Da,

Sc #>= Sa + Da,

Sd #>= Sb + Db,

Sd #>= Sc + Dc,Se #>= Sb + Db,

Sf #>= Sd + Dd,

Sg #>= Se + De,

Sg #>= Sf + Df,indomain(Sg),

labeling([Sa,Sb,Sc,Sd,Se,Sf,Sg]),



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Project planning results

Simple project example

B E

013

A G

3

4 2

23 7 9

C D F

3 2 4



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Disjunctive resources

Constraint HandledA task uses resource exclusively during its duration

Modelling Constraint as Choices (too weak)

on ona ropaga on oo wea

Cumulative constraint

Alternative Modelling Job sequencing



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Cumulative formulation

Express disjunctive with one cumulative

constraint a, ,.. , a, ,... , , ,... , ,..

Active use of the constraint

Recognition of capacity problems available time does not allow all jobs to be scheduled

bottleneck periods when some jobs have to be scheduled

possibi lity to link directly with project end via cumulativeconstraint parameter

Limited interaction with precedenceconstraints



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Alternative formulation

Express disjunctive as sequence of tasks on

a machine

variable Task_x, value startdate

variable first task, value task_x

Classical formulation

Problem when combining with other (e.g.



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-

Example



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Jobs and tasks

Job Task Duration Machine

J1 T11 5 M1

J2 T21 3 M2

J2 T23 4 M2

J4 T41 7 M1



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Job shop program

End = 100,

[T11,T12,T21,T22,T23,T31,T41] :: 0..End,[E1,E2,E3,E4] :: 0..End,

[D11,D12,D21,D22,D23,D31,D41] = [5,7,3,5,4,5,7],

T12 #>= T11+D11, T22 #>= T21+D21, T23 #>= T22+D22,E1 #= T12+D12, E2 #= T23+D23,

E3 #= T31+D31, E4 #= T41+D41,

cumulative([T11,T22,T41],[D11,D22,D41],[1,1,1],1),

cumulative([T12,T21,T23,T31],[D12,D21,D23,D31],[1,1,1,1],1),

min_max(labeling([T11,T12,T21,T22,T23,T31,T41]),

[E1,E2,E3,E4]),



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Job shop solution

Minimize Cmax

Optimal solution 19

T21 T31 T12 T23

5 10 15 20 250



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Cumulative machines

Constraint HandledAlternative Machines

Availability Profiles

Generalization of disjunctive case

Constraint Modelling sets of 0/1 variables

cumulative formulation

Relation to Machine Assignment



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Modelling

Required for each task start date Si

resource consumption Ri

Global information vera resource m

x :: .. ,

cumulative([Sa, Sb,...],[Da, Db,...],[Ra, Rb,...],Rx,...)



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Multiple machines

End = 100,

[T11,T12,T21,T22,T23,T31,T41] :: 0..End,

[E1,E2,E3,E4] :: 0..End,

D11 D12 D21 D22 D23 D31 D41 = 5 7 3 5 4 5 7

T12 #>= T11+D11, T22 #>= T21+D21, T23 #>= T22+D22,

E1 #= T12+D12, E2 #= T23+D23,

E3 #= T31+D31, E4 #= T41+D41,



min_max(labeling([T11,T12,T21,T22,T23,T31,T41]), [E1,E2,E3,E4]),

Example



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Multiple machines

Two copies of machine 2

Cumulative solution - not assignment

Optimal solution cost 17

5 10 15 20 250

T21

T31 T12 T23

5 10 15 20 250



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Manpower use

Job Task Duration Machine Manpowe

J1 T11 5 M1 2

J1 T12 7 M2 3

J2 T21 3 M2 4

J2 T22 5 M1 3

J2 T23 4 M2 3

J3 T31 5 M2 2

J4 T41 7 M1 4



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Manpower model

End = 100,

[T11,T12,T21,T22,T23,T31,T41] :: 0..End,

[E1,E2,E3,E4] :: 0..End,

[D11,D12,D21,D22,D23,D31,D41] = [5,7,3,5,4,5,7],

Manpower = 6,

[U11,U12,U21,U22,U23,U31,U41] = [2,3,4,3,3,2,4],T12 #>= T11+D11,T22 #>= T21+D21,T23 #>= T22+D22,

E1 #= T12+D12,E2 #= T23+D23,E3 #= T31+D31,E4 #= T41+D41,



Man :: 0..Man ower,

cumulative([T11,T12,T21,T22,T23,T31,T41],[D11,D12,D21,D22,D23,D31,D41],

U11 U12 U21 U22 U23 U31 U41 Manmin_max(labeling([T11,T12,T21,T22,T23,T31,T41]),

[E1,E2,E3,E4]),



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Manpower result

Optimum cost 21

Manpower use 6

5 10 15 20 250

T21 T31T12

T23

5 10 15 20 250



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Unavailability

Handling of availabil ity profiles introduction of fixed dummy tasks

-

ummy as s

profile

t



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Relation to assignment

The cumulative problem states a necessary

condit ion for solving the assignment problem

Assignment problem is trivial, if machine

Typically, assignment can be solved easily, ifa few resource switches are allowed

If assignment is very important, otherconstraints must be added



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Consumable resources

Constraint Handled

Raw Materials

Stocks

Receivings

Orders

Idea an e consuma e resources as a spec a ype o

renewable resource Time points become periods

from t ime point to end



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Producer constraint

Example: Finished Products

Demand given for fixed time point and fixed

time points

quantity required

Determine the dates when to produce tasks inorder to satisfy demand

enddate (fixed) quantity produced



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Consumer constraint

Example: Raw Materials

Supply given for fixed time point and fixed

time points

quantity required

Determine the dates when to produce tasks inorder to respect supply

enddate (fixed) quantity consumed



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Producer consumer

Example: Intermediate Products

Material produced by Producer tasks en a es

(fixed) quantity

Material consumed by Consumer tasks start dates

(fixed) quantity

Constraint: no negatives

stoc must e pro uce e ore t s consume



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Producer consumer (2)

Producer events are represented by

tasks from overall start to end of task

tasks from start of task to overall end

Resource limit is set as sum of all producedquantit ies + initial stock



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Cumulative Modelling

graphical representation of producer/

consumer constraints

Supply

Limit

u

(variable)Supply

(variable)

Demand

Stock

t



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Product requirements

ro uce y onsume y

J1 60 6 60

J2 70 12 40

J3 50 21 100

J4 40

0 20

Example



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Producer model

End = 100,

Stock = 20,

[Q1,Q2,Q3,Q4] = [60,70,50,40],

Limit :: 0..1000,

Limit #


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Stock level

-60 -40 -100

60

5 10 15 20 250

40

Manpower use

5 10 15 20 250

T21T31 T12

T23

5 10 15 20 250



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Stock level

-60 -40 -100

60

5 10 15 20 250

40

Demand

J4

J2J1

5 10 15 20 250

DemandJ3

Stock

T21T31 T12

T23

5 10 15 20 250


E t i


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Extensions

Variable resource consumption/production

variable height of rectangles + dummy tasks

maximum negative stock at any time

total negative stock*time amount

use o n erme a e resource m n cumu a ve

Splitting resource use consumer tasks do not need all material in the beginning

producer tasks produce some material before the end

pipelining operations

jit production

Safety margins


Li it d t k l l


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Limited stock levels

Problem

only l imited amount of stocks can be stored

consumption

Solution: another call to cumulative

ea: inverse roles of producer and consumer

add maximum stock level


M hi h i


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Machine choice

tasks may run on different machines, but may not overlap

each other each machine is running at most one task at a time

sets of possible machines for different tasks may overlappartially

Constraint Handled:Alternative, not equivalent machines

different speed of machines (different capabil ities)

machine de endent duration of tasks

flexible work cells


M d lli


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Modelling

Additional degree of freedom

dramatic increase of search space

Machine assignment variable Mi

element constraint (functional dependencies) link between machine and duration

element(Mi, [...], p i)

decide machine assignment beforehand load balancing


Non o erlapping constraint


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Non overlapping constraint

Mathematical Formulation

for two tasks i and j we have=

t i >= tj+pj or

tj >= t i+p i

Conditional Propagation

far too weak

ec ang e acemen

diffn constraint Redundant cumulative constraint

e axat on p acement -> cumu at ve


Diffn model


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Diffn model

Constraint expressed as 2D placement

X-dimension = time Y-dimension = resource

tasks are rectangles with height 1

task may be assigned to dif ferent resource

Duration (variable)

task ma run at different timeResource

Start


Redundant cumulative


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Redundant cumulative

Relaxation

each task has resource use one

resource limit = number of machines

cumulative constraint on resource use

duration of tasks variable

good for full schedules

update of time periods of remaining tasks

Redundant constraint adds reasoning power


Machine speed


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Machine speed

Task M1 M2 M3

T11 5 3 3T12 7 7 7

T21 4 3 3

T22 5 7 7

T23 4 4 4

T31 6 5 5

Example


Machine choice model


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Machine choice model

[M11,M12,M21,M22,M23,M31,M41] :: 1..3,

machine_duration([M11,M12,M21,M22,M23,M31,M41],[D11,D12,D21,D22,D23,D31,D41],

[[5,3,3],[7,7,7],[4,3,3],[5,7,7],[4,4,4],[6,5,5],[7,6,6]]),

, , , , , , , , , , , ,[T22,M22,D22,1],[T23,M23,D23,1], diffn

constraint

min_max(labeling([T11,T12,T21,T22,T23,T31,T41,M11,M12,M21,M22,M23,M31,M41]),[E1,E2,E3,E4]),


-60 -40 -100

70


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Stock level

70

5 10 15 20 250

40 50 60

Manpower use

5 10 15 20 250

T11

T41

T21 T12

5 10 15 20 250


Sequence dependent setup


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Sequence dependent setup

Constraint Handled:

Tooling

Adjustment Periods

Constraint Modelling

yc e cons ra n cu pro ems Diffn constraint (large problems)

exclusion markers, not discussed here


Setup cycles


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Setup cycles

All tasks on one disjunctive resource form a

cycle

successor)

cycle with minimum weight = minimum setup

T1 T2 T3R1

N1 N3N2

setup1,2 setup2,3

0

R1


Setup relaxation


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Setup relaxation

Problem:

setup depends not just on one task; but on two

lower bound often zero

Relaxation of problem:

gnore se up mes no s gn can e.g. 5 min setup for 2 hour job

Include setup in duration of task if not sequence

extend duration of tasks

Over-estimate setup by worst case assumption


setup


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setup

Constraint Handled:

Setup on between tasks which are assigned to the samemachine

Combination with machine choice

Modelling

multiple cycles in one constraint

generalisation of one machine case


Preemption


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Preemption

Constraint Handled:

Breaks

Weekend

Machine dependent calendars

- Constraint Modelling

Preemption by Downtime

e emen cons ra n s

Inequalities Preemption by task switch

ara gm sw tc


Cost


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Cost

Cost reporting

A cost value that should be computed for a solution

Cost constraintA cost which is handled as a constraint in the search

p a e ns e e cons ra n propaga on

Cost approximationA constraint which is not a cost of the application, but

added to improve pruning in the search

Different cost constraints have differentsearch behavior


Cost: maximum end date


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Cost: maximum end date

Typical: Overall Project End

related to machine utilization

explicit end task

inequalities

genera oma n m

Good cost estimation by constraints


Cost: sum of lateness


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Cost: sum of lateness

Scheduling with due-dates

Expressed by compu a on o a eness

constraint approximation

Ti >= ti+p i - d i

ar me c cons ra n on we g e sum

Reasonable cost estimation by constraints added estimate of individual lateness

Poor propagation from cost limit individual lateness only limited very late in search tree


Cost: maximum lateness


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Cost: maximum lateness

Scheduling with due-dates

Expressed by compu a on o a eness

constraint approximation

Ti >= ti+p i - d i

nequa es o eac a eness

Good cost estimation by constraints estimate of individual lateness

Good propagation from cost limit individual lateness immediately


Cost: preferred start date


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Cos p e e ed s a da e

JIT scheduling

stock minimization

Expressed by two variables lateness and earlyness

eac approx ma e y nequa y

Restricts domains to (small) intervals good propagation

possibility of failure


Cost: resource usage


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g

Optimizing factory throughput

Machine choice Expressed by

Sum of duration p i

Use as a strategy run task on fastest machine


Cost: stock level


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Stock

Stock Level

Stock Volume

Negatives

cumulative constraints

Good estimates intermediate resource levels in cumulative

Poor propagation from constraint limit

constraint too restrictive for remaining schedule


Strategies


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g

Need for strategies

constraint solver not complete (complexity)

use of domain knowledge (intelligence)

Heuristic solutions

n one or ew so u ons w a goo qua y strategy used for finding good solution

Optimization procedures control search procedure to prove optimality

strategy used to limit search tree


Heuristics


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Degrees of Freedom

Inside application domain

relaxation

decomposition

variable assignment

search strategy


Co-operation


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p

Constraints and Heuristics should work

together

next choice

Dynamic heuristics

choice re-evaluated at each step

Better than static heuristics

based only on static picture

Without constraints heuristic must simulate its effect


Assignment strategy


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Choose variable

Depending on domains of variables

Depending on heuristics

Choose value

se o e res r c e oma n Use of heuristics

Alternative: Choose set of alternative constraints

Choose among alternatives


Labeling strategy


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Two possible extremes; mixed forms possible

Value choice e erm n s c c o ce o var a e

Non-deterministic choice of value

Variable choice Non-deterministic choice of variable

Deterministic choice of value

Deterministic choice of variable

Deterministic choice of value


Value choice


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Often used for CSP problems

First fail strategy

Supported by primitives labeling

e e e indomain

Typical codelabeling([]).

labeling([H|T]):-delete(X,[H|T],R,0,first_fail),

indomain(X),

labeling(R).


Variable choice


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Often used in OR methods

Left-most strategy

all other tasks do not start earlier

on backtracking, try another variable

Simple example delete/3

delete(X,[X|R],R).

e e e , , :-

delete(X,R,S). Support by primitives

delete/5

domain_info


Redundant constraints


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Add constraint to improve reasoning

Consequence of incomplete solver

Add constraints to restrict admissiblesolutions

reng en cons ra n s Find only one solution, not all

Relaxation techniques


Optimization


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Search trees

Search space

Sub tree

Choice points

Improved

Solutions

Initial

Solution


Importance of lower bounds


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Cut parts of search tree

Complete enumeration impossible

cost The better the cost estimate, the earl ier we can prune the

Depends on cost function and constraints The pruning of the constraints may help to find good cost

Avoid additive costs No direct influence between local decision and global

cos


Partial optimization


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Not always necessary to explore completesearch tree

Imprecise data

Allow limited optimization around good

Consider only one or two alternatives at each choicepoint

Consider decomposition to optimize subproblems


Methodology summary


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Modelling of scheduling problems withconstraints

cumulative

diffn

cyc e Mixing of constraints

toolbox

incremental problem Modelling

Role of strategies

optimisation


Part III: Applications


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Benchmarks and their limits

ATLAS Other examples


Benchmark problems


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Bridge Project planning with resource constraints

Difficul t job-shop

Patterson 110 cumulative problems

Alvarez


Limits of benchmarks


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Pure problems Non incremental techniques

Number of problem instances Over-specializing methods

eur s cs oo par cu ar Performance measures

Number of backtrack points

Run time

Problem sizes

2


ATLAS


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Scheduling system in chemical factory herbicide manufacturing and packaging

Live system first phase operational since July 93

secon p ase opera ona s nce arc Developed jointly by

Beyers & Partners

COSYTEC

Multi-user PC/UNIX system

co-operative behavior


The process


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F1L1

F2

L3

L2

F3 L4Area

F4

F5L6

Formulation Storage Canning Outside

Canning

Container

Storage


Batch part


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Herbicide formulation

Main constraints eren pro uc on mes or resources

typical 4-8 hours

Batch sizes between min and max limits

e.g. o ers no smaller batches allowed

Product dependent setup

very ong compare o pro uc on me

Limited choice of resources

Very l imited storage capacity

typ ca atc

Exceptional use of auxiliary storage


Packaging part


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Fill ing of product into containers of differentsizes .

Packaging lines vary in capabil ity Speed

a er a an e Setup

Manpower required

Outside packaging scheduled manually

But taken into account for stocks and materials Operation can not be automated


The system


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Connection to other systems Data download/upload from/to local and remote

mainframes

End-user connection via PCs

Data base

ORACLE SQL standard

Forms/reports facilities

u -user oo

Multiple schedulers for parts of production Inventory control

Production

Multiple scenarios; one schedule


System integration


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- IBM - VMSSAS

X25

HP 9000

Oracle

PCEthernet


Graphical interface

Diff t i i t d t


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Different views into data Graphical/textual display of information

Moving tasks in t ime and on resources

Checks on constraints

Control on manual modificationsAllows constraint violations

User can modify/change schedule at any time


Views into data

Diff t i f d t


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Different views of data Gantt

Orders

Finished Products

Raw Material


Constraint solver

C t i t h dl d


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Constraints handled Machine choice

Manpower limits

Component consumer constraints

Producer/consumer constraints for intermediate products

Production planning for batch part

Optimum batch sizing

Limited stock levels


Machine choice

Tasks can be run on different lines


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Tasks can be run on different lines Different production rates

Different setups

Heuristic choice of preference to fastest line

Offset by need to keep l ines busy


Setup

Setup depends on characteristics of products


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Setup depends on characteristics of products Bottles

Formulated product

Setup varies very much

o re axa on poss e Choice influences overall throughput

Conflict with due dates

Uses strategy to minimize setup close to (theoretical) optimum


Components

Component availabil ity required for


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Component availabil ity required forproduction

Frequent changes in BOM BOM local to task

Constrains schedule up to lead time ofcomponent Typical up to three weeks

Delivery of material known up to lead time Delivery

Material orders


Components (2)

Tasks consume stocks on certain time points


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Tasks consume stocks on certain time points Not all material must be available at start date

Large increase of constraint size

Consumer constraint on consumable

Simple forms can be replaced by inequalities


Finished products

Known delivery dates of finished products


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Known delivery dates of finished products Orders of fixed quantity

Data reconci liation problem

Finished product available before end of

tasks Handled on a per day /shift basis

Increases problem size

Important to minimize stock levels

Producer constraint with cumulative


Formulated product

Producer consumer constraint between


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Producer consumer constraint between Batch formulation (producer)

Fixed parts: stocks, transfers, material orders etc

Limited overlap possible a c es are ava a e a en o ormu a on

canning tasks do not need all product at start date

Batch size dependent on actual use


Manpower

Manpower limits given from database


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Manpower limits given from database Presented on per shif t basis

of tasks Depends on canning line chosen

u oma e nes

Manual lines

Additional manpower limits on number oflines in operation

Supervisors/technicians


Production planning

Tasks for canning lines given from database


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Tasks for canning lines given from database Related to orders/sales forecasts

Tasks for formulation generated by system Number of batches determined by system

c e u e n ma n sc e u er

Separate phase to tune batch sizes


Multiple scenarios

Possibi lity to store/recall/delete multiple


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y pscenarios

Exchange of proposed solutions betweenoperators

What-if evaluation Comparison manual/automated scheduler


Quality indicators

Orders too late


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Tasks too late Tasks too early

Stock level

Production rate

Setup periods/duration/manpower

Cleaning time/ resources


ATLAS - Summary

End-user system


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y Direct manipulation

Changes/strategies expressed via interface

Complex scheduling problem ea - e, no pure pro em

Interaction/conflicts between constraints

Incremental expression with constraints

Significant development effort

Integration Data base reporting

Graphical interface

Problem solver


Other applications

HIT (ICL)


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Scheduling/assignment for container harbor

Planning/scheduling of mail sorting

PLANE (Dassault Aviation)

Production line scheduling

Workshop scheduler (Dassault Aviation)

Saveplan (Sligos) MRP scheduling package

PROTOS (BIM, Hoechst, IBM, Sandoz) Scheduling project for chemical industry


Applications (2)

CPLAN (COSYTEC)


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Toolbox for resource restricted project planning

Scheduling planning/simulation for refineries

APACHE (COSYTEC)

Assignment for airport stand allocation

PILOT (SAS Data/COSYTEC)

EVA (EDF/GIST) Transport schedule for nuclear fuel

TACT (COSYTEC) Transport schedule in food industry


Part IV: to probe deeper

Comparison


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Pitfalls Part of the development process

References


ompar son

Alternative MethodsOR t h i


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OR techniques

Heuristic approach

Iterative Improvement

Evaluation Criteria

Expertise

Efficiency


When to use constraints Hard Problems


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Admissible solutions are not obvious

Strong interaction of different constraints

Local Costs os s are expresse oca y, no g o a y

Lower bound estimated by constraints

Strategies are important User knowledge can be included

No blind search for an optimum


Pitfalls Modeling Everything


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Constraints must constrain

Soft/Hard Constraints Soft constraints do not help

ar m s e n so ones

Optimization with imprecise data Influence on problem size

Theoretical vs practical optimum

KISS principle applies ,


Relative importance of solver Percentage of design

S l t b t k i t t d i d i


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Solver must be taken into account during design

Percentage of development Often small compared to integration work

uccess no guaran ee

Prototyping

Experiments early in development cycle

e a ve y g r s compare o o er eve opmen

Importance to overall acceptance Solver alone is not enough

Without solver the rest is not useful


References References for the areas

C t i t th d


152/152

Constraint methods

Applications of CLP

CLP systems

Other constraint tutorials

Presentations at PAP 95

Documents

Scheduling and Planning With Constraint Logic Programming