lesson 2 advanced scheduling topics - ai @ uniboai.unibo.it/system/files/u11/lesson2 v3_1.pdf ·...

Lesson 2Advanced Scheduling Topics

DEIS, University of Bologna

Outline

Main Objective: introduce some more advanced techniques

In particular, an overview of:§ Part 1: Scheduling via Precedence Constraint Posting§ Part 2: Large Neighborhood Search for Scheduling Problems

Lesson 2, Part 1:Advanced Scheduling Topics


Precedence Constraint Posting

A Different Perspective

2

t

A4A1

Let’s schedule a single pair of activitiesThere are two things we may do:

A2

A3

A5 A6 A7

1/2

1/1

1/2 3/2 2/1

A1

A4

2/1

1/2


2

t

A1

§ Set s1 = 0 and s4 = 2

A1

A4

2/1

1/2

A4



2

t

§ Set s4 = 0 and s1 = 1

A1

A4

2/1

1/2

A4A1

§ Set s1 = 0 and s4 = 2


In other words:§ Run A1 before A4§ Run A4 before A1


Let’s schedule a single pair of activitiesThere are two things we may do:§ Set s1 = 0 and s4 = 2§ Set s4 = 0 and s1 = 4

A1

A4

2/1

1/2In other words:§ Run A1 before A4§ Run A4 before A1

Scheduling choices can be stated as ordering decisionsThis idea is called Precedence Constraint Posting [10bis]§ This captures the problem core more concisely!§ Can be used to formulate a model

Disjuctive RCPSP Model

Let’s start from the previous one

min z = max

ai2Aei

ei = si + di 8ai 2 A

ei sj 8(ai, aj) 2 E

cumulative([si], [di], [rqi,k], ck) 8rk 2 R

si 2 [0..eoh] 8ai 2 A

ei 2 [0..eoh] 8ai 2 A



min z = max

ai2Aei


ei sj 8(ai, aj) 2 E


si 2 [0..eoh] 8ai 2 A

ei 2 [0..eoh] 8ai 2 A

8(ai, aj) /2 Ebfi,j 2 {0, 1}“Before” variables

bfi,j = 1 , ei sj



min z = max

ai2Aei


ei sj 8(ai, aj) 2 E


si 2 [0..eoh] 8ai 2 A

ei 2 [0..eoh] 8ai 2 A

8(ai, aj) /2 Ebfi,j 2 {0, 1}

8(ai, aj) /2 E“Before”

constraints

Why Precedence Constraint Posting?

Why should one use this approach?


§ The decision variables in the previous model are:

§ The decision variables in the new model are:

si 2 [0..eoh] 8ai 2 A

ei 2 [0..eoh] 8ai 2 A

8(ai, aj) /2 Ebfi,j 2 {0, 1}

Reason #1: the domains are not large at all!


§ Assume activity durations are uncertain or that thy depend on some unassigned variable

§ Ordering decision are robust against duration variability

Reason #2: PCP guarantees resource consistency even with uncertain or variable durations

dmin

A4A1

dmaxdmin dmax

A Big Drawback!

bfi,j = 1 , ei sj

min z = max

ai2Aei


ei sj 8(ai, aj) 2 E


si 2 [0..eoh] 8ai 2 A

ei 2 [0..eoh] 8ai 2 A

8(ai, aj) /2 Ebfi,j 2 {0, 1}

A (very) relevant issue: since§ We are not assigning start times§ Cumulative propagation is incompleteAssigning bfi,j does not guarantee resource consistency

Does not guarantee

consistency

bfi,j + bfj,i = 1ai, aj

(ai, aj) /2 E | 9rk 2 R : rqi,k = rqj,k = 1

...and a Welcome Exception

Assume we have only unary resources, then:§ Every pair of using the same resource must be sequenced§ Hence we must have:§ For every:

This model (the key idea of it) is common in Job Shop SchedulingIn [8], the authors obtained pretty nice results by using:§ Very simple propagation§ dom wdeg1 for variable selection (a general purpose heuristic)§ Restarts (+ a bit of no-good recording)§ The best solution found so far as a value selection heuristic

1: Domain over Weighted Degree (it’s an adaptive heuristic)

Generalizing

What about non-unary resources?

2

t

§ A, B, C cause an over-usage in case of overlapping execution

C

B

A

Generalizing


2

t


§ Adding a single precedence constraint is enough to prevent the conflict

CB

A

Generalizing


2

t


§ Adding a single precedence constraint is enough to prevent the conflict

CB

A

This is called a Minimal Critical Set [9,10]

Generalizing

bfi,j = 1 , ei sj

This is for unary resourcesmin z = max

ai2Aei


ei sj 8(ai, aj) 2 E


si 2 [0..eoh] 8ai 2 A

ei 2 [0..eoh] 8ai 2 A

8(ai, aj) /2 Ebfi,j 2 {0, 1}

8(ai, aj) /2 E | 9rk 2 R : rqi,k = rqj,k = 1bfi,j + bfj,i = 1

X

ai2MCS

bfi,j >= 1 8MCS in the graph

Generalizing

bfi,j = 1 , ei sj

And this is for non-unary resourcesmin z = max

ai2Aei

ei = si + di 8ai 2 Aei sj 8(ai, aj) 2 E


si 2 [0..eoh] 8ai 2 A

ei 2 [0..eoh] 8ai 2 A8(ai, aj) /2 Ebfi,j 2 {0, 1}

X

ai2MCS

bfi,j >= 1 8MCS in the graph

Trouble to Come

On a project graph, a Minimal Critical Set is:§ A set of non precedence connected activities§ With the two properties mentioned

In the worst case, MCS may be in exponential number!(and the worst case does happen)

Typically: detect and resolver MCS at search time

An MCS Based Search Strategy

A possible search strategy [11]:

found?

Select an MCS

select a pairai, aj

Solutionno

yes

post .

branch

ei sj post .ei > sj

on backtracking



found?

Select an MCS

select a pairai, aj

Solutionno

yes

post .

branch

ei sj post .ei > sj

on backtracking

No MCS = no possible conflictsThe precedence constraints guarantee resource consistency

This is called a Partial Order Schedule



found?

Select an MCS

select a pairai, aj

Solutionno

yes

post .

branch

ei sj post .ei > sj

on backtracking

The two main steps

Selecting MCS and Activity Pairs

Selecting ai, aj pairsThere are some heuristics:§ Preserved space§ Slack- and Texture-based

Selecting an MCSA few available methods:§ Enumeration [11]

Selecting MCS via Enumeration

Enumerating MCS§ Sort activities by decreasing requirement§ Enumerate the subset

A1 A2

A3

A5 A6

A4

A7

2/1

1/2

1/1

1/2

1/2 3/2 2/1

ck = 2

A2A4

A7A1A3 A5 A6


Enumerating MCS§ Sort activities by decreasing requirement§ Enumerate the subset§ Skip precedence connected activities

A1 A2

A3

A5 A6

A4

A7

2/1

1/2

1/1

1/2

1/2 3/2 2/1

ck = 2

A2A4

A7A1A3 A5 A6



§ When the capacity is exceeded, an MCS has been found

A1 A2

A3

A5 A6

A4

A7

2/1

1/2

1/1

1/2

1/2 3/2 2/1

ck = 2

A2A4

A7A1A3 A5 A6

MCS found




§ Keep the best MCS according to ranking heuristic (size, weight, preserved space...)

A1 A2

A3

A5 A6

A4

A7

2/1

1/2

1/1

1/2

1/2 3/2 2/1

ck = 2

A2A4

A7A1A3 A5 A6

best = A3, A4




§ Keep the best MCS according to ranking heuristic (size, weight, preserved space...)

A1 A2

A3

A5 A6

A4

A7

2/1

1/2

1/1

1/2

1/2 3/2 2/1

ck = 2

A2A4

A7A1A3 A5 A6

best = A3, A4MCS found




§ Keep the best MCS according to ranking heuristic (size, preserved space...)

A1 A2

A3

A5 A6

A4

A7

2/1

1/2

1/1

1/2

1/2 3/2 2/1

ck = 3

A2A4

A7A1A3 A5 A6

best = A3, A4MCS found...

PRO: we find the “best” MCSCON: the process may take exponential time!

Selecting MCS and Activity Pairs


Selecting an MCSA few available methods:§ Enumeration [11]§ Reduction of the Max Weight Independent Set [12]


§ An MCS is an Independent Set on the Project Graph

§ It is possible to find the Independent Set of maximum weight in polynomial time!

A1 A2

A3

A5 A6

A4

A7

2/1

1/2

1/1

1/2

1/2 3/2 2/1




Finding the Max Weight Independent Set§ Split nodes and turn requirements into

minimum capacity labels

A1 A2

A3

A5 A6

A4

A7

1

2

1

2

2 2 1

A1 A2

A3 A4

A5 A6 A7




A1 A2

A3

A5 A6

A4

A7

1

2

1

2

2 2 1

A1 A2

A3 A4

A5 A6 A7

S

T


minimum capacity labels§ Add fake Source and Target nodes




A1 A2

A3

A5 A6

A4

A7

1

2

1

2

2 2 1

A1 A2

A3 A4

A5 A6 A7

S

T

4

2 2

2


minimum capacity labels§ Add fake Source and Target nodes§ Route minimum flow from S to T, such that

all minimum capacity labels are covered§ e.g. use inverse Fold Fulkerson




A1 A2

A3

A5 A6

A4

A7

1

2

1

2

2 2 1

A1 A2

A3 A4

A5 A6 A7

S

T


minimum capacity labels§ Add fake Source and Target nodes§ Route minimum flow from S to T, such that

all minimum capacity labels are covered§ e.g. use inverse Fold Fulkerson

§ Sets of saturated arcs separating S and T (i.e. S/T cut) identify the IS with max weight


Here there is a single max weight IS

A1 A2

A3

A5 A6

A4

A7

1

2

1

2

2 2 1

A1 A2

A3 A4

A5 A6 A7

S

T

A4 A5 A6

§ This is not minimal!§ But can be minimized§ Method 1: remove activities according

to a heuristic (size, preserved space...)§ Method 2: sampling (bounded

enumeration)


Here there is a single max weight IS

A1 A2

A3

A5 A6

A4

A7

1

2

1

2

2 2 1

A1 A2

A3 A4

A5 A6 A7

S

T

A4 A5 A6

§ This is not minimal!§ But can be minimized§ Method 1: remove activities according

to a heuristic (size, preserved space...)§ Method 2: sampling (bounded

enumeration)

PRO: extremely efficient!Method 1 also has no exponential worst case

CON: the MCS is not necessarily the “best” one



Selecting an MCSA few available methods:§ Enumeration [11]§ Reduction of the Max Weight Independent Set [12]§ Sampling from a Maximum Usage Envelope [13]


A max usage envelope defines the max resource usage at each time instant A1 A2

A3

A5 A6

A4

A7

2/1

1/2

1/1

1/2

1/2 3/2 2/1

Finding the max usage envelope§ Define two events per activity,

corresponding to ESTi and LETi

[2,9]/[4,11]

A1: [0,5]/[2,7]A2: [0,7]/[1,8]A3: [2,7]/[3,8]A4: [1,8]/[2,9]

A6: [3,8]/[6,11]A5: [3,10]/[4,11]

A7: [2,9]/[4,11]



§ Extract the corresponding time points


A1 A2

A3

A5 A6

A4

A7

2/1

1/2

1/1

1/2

1/2 3/2 2/1

0 7 821 93 11[2,9]/[4,11]

2

t

A1: [0,5]/[2,7]A2: [0,7]/[1,8]A3: [2,7]/[3,8]A4: [1,8]/[2,9]

A6: [3,8]/[6,11]A5: [3,10]/[4,11]

A7: [2,9]/[4,11]



§ Extract the corresponding time points


0 7 821 93 11[2,9]/[4,11]

2

t

A1: [0,5]/[2,7]A2: [0,7]/[1,8]A3: [2,7]/[3,8]A4: [1,8]/[2,9]

A6: [3,8]/[6,11]A5: [3,10]/[4,11]

A7: [2,9]/[4,11]

A1 A2

A3

A5 A6

A4

A7

1

2

1

2

2 2

A1 A2

A3 A4

A5 A6 A7

S

T

§ Transform the graph as before


0 7 821 93 11[2,9]/[4,11]

2

t

A1: [0,5]/[2,7]A2: [0,7]/[1,8]A3: [2,7]/[3,8]A4: [1,8]/[2,9]

A6: [3,8]/[6,11]A5: [3,10]/[4,11]

A7: [2,9]/[4,11]

A1 A2

A3

A5 A6

A4

A7

1

2

1

2

2 2

A1 A2

A3 A4

A5 A6 A7

S

T

Finding the max usage envelope§ ...§ For each time point t:§ If LETi ≤ t or ESTi > t, remove

corresponding nodes


0 7 821 93 11[2,9]/[4,11]

2

t

A1: [0,5]/[2,7]A2: [0,7]/[1,8]A3: [2,7]/[3,8]A4: [1,8]/[2,9]

A6: [3,8]/[6,11]A5: [3,10]/[4,11]

A7: [2,9]/[4,11]


corresponding nodes§ Identify the max weight IS via flow

minimization

A1 A2

1A1 A2

S

T

1 1

A1,A2

A1 A2

A3

A5 A6

A4

A7

1

2

1

2

2 2 1

A1 A2

A3 A4

A5 A6 A7

S

T


0 7 821 93 11[2,9]/[4,11]

2

t

A1: [0,5]/[2,7]A2: [0,7]/[1,8]A3: [2,7]/[3,8]A4: [1,8]/[2,9]

A6: [3,8]/[6,11]A5: [3,10]/[4,11]

A7: [2,9]/[4,11]



minimization

A1,A2


0 7 821 93 11[2,9]/[4,11]

2

t

A1: [0,5]/[2,7]A2: [0,7]/[1,8]A3: [2,7]/[3,8]A4: [1,8]/[2,9]

A6: [3,8]/[6,11]A5: [3,10]/[4,11]

A7: [2,9]/[4,11]



minimization

A1 A2

A4

1 1

2

A1 A2

A4

S

T

A1,A2

A1,A4

1 2

2

t


0 7 821 93 11[2,9]/[4,11]

A1: [0,5]/[2,7]A2: [0,7]/[1,8]A3: [2,7]/[3,8]A4: [1,8]/[2,9]

A6: [3,8]/[6,11]A5: [3,10]/[4,11]

A7: [2,9]/[4,11]

Finding the max usage envelope§ ...

A1,A2

A1 A2

A3

A5 A6

A4

A7

1

2

1

2

2 2 1

A1 A2

A3 A4

A5 A6 A7

S

T

A1,A4

A3,A4

A3,A4 A3,A4

A4,A5

A5,A7

2

t


0 7 821 93 11[2,9]/[4,11]

A1: [0,5]/[2,7]A2: [0,7]/[1,8]A3: [2,7]/[3,8]A4: [1,8]/[2,9]

A6: [3,8]/[6,11]A5: [3,10]/[4,11]

A7: [2,9]/[4,11]

Finding the max usage envelope§ ...§ Peaks (may) correspond to a CS§ Obtain an MCS from each CS via

minimization (removal or sampling)

A1,A2

A1 A2

A3

A5 A6

A4

A7

1

2

1

2

2 2 1

A1 A2

A3 A4

A5 A6 A7

S

T

A1,A4

A3,A4

A3,A4 A3,A4

A4,A5

A5,A7

A pretty good trade-off between the previous two

0.25%0.5%

0.75%

1%

0%5%10%15%20%25%30%

0.2%0.5%

0.7%1%

0.25%0.5%

0.75%

1%

0%5%10%15%20%25%30%

0.2%0.5%

0.7%1%

0.25%0.5%

0.75%

1%

0%5%10%15%20%25%30%

0.2%0.5%

0.7%1%

Results for the J60 PSPLIB instances

[2,9]/[4,11]

Number of optimally solved instances in 10 mins

num of req. res.

ENUM

ENVELOPE (sampling)

rqi,k values

0.25%0.5%

0.75%

1%

0%5%10%15%20%25%30%

0.2%0.5%

0.7%1%

0.25%0.5%

0.75%

1%

0%5%10%15%20%25%30%

0.2%0.5%

0.7%1%

0.25%0.5%

0.75%

1%

0%5%10%15%20%25%30%

0.2%0.5%

0.7%1%


[2,9]/[4,11]

ENUM

The envelope method is the best here

MAX IS (removal)

ENVELOPE (sampling)

0.25%0.5%

0.75%

1%

0%5%10%15%20%25%30%

0.2%0.5%

0.7%1%

0.25%0.5%

0.75%

1%

0%5%10%15%20%25%30%

0.2%0.5%

0.7%1%

0.25%0.5%

0.75%

1%

0%5%10%15%20%25%30%

0.2%0.5%

0.7%1%


[2,9]/[4,11]

ENUM

The max weight IS method seems to have the best scalability

ENVELOPE (sampling)

MAX IS (removal)



Robust Search Startegies

maxai2A si(�) + diPai2A max(0, si(�) + di � dli)

Pai2A max(0, dli � si(�)� di)


Scheduling Problem come in many variants§ Unary, cumulative and state resources§ Minimal and maximal time lags, setup times§ Different objective functions, e.g.:

makespan:

weighted tardiness:

weighted earliness:

Most of the methods we have seenwork well only for specific cases





makespan:

weighted tardiness:

weighted earliness:


Bad for Schedule or Postpone





makespan:

weighted tardiness:

weighted earliness:


Poor propagation





makespan:

weighted tardiness:

weighted earliness:

A solution: use specialized methodsBut this is very inconvenient!

Poor propagation

Self-Adaptive Large Neighborhood Search

Large Neighborhood Search § A heuristic approach§ Iteratively try to improve an

initial solution via CP§ Two main steps§ Works for all the RCPSP

variants seen so far :-)

Self-Adaptive Large Neighborhood Search [14] has been recently applied with good results to a diverse set of benchmarks

Find a solution

Relax a fragment

Complete therelaxed solution

improved?

Change incumbent solution

no yes

SA-LNS Fragment Relaxation

2

t

A2

A1A3

Assume this is the initial solution:

A4 A5 A6A7


Turn the start time assignment into a Partial Order Schedule§ Methods available from [13,15] for many resource types

2

t

A2

A1A3 A4 A5 A6

A7


Turn the start time assignment into a Partial Order Schedule§ Methods available from [13,15] for many resource types

2

t

A2

A1A3

A4

A5

A6

A7Original

Added


Select a subset of activities§ Choice guided by a randomized relaxation heuristic

2

t

A2

A1A3

A4

A5

A6

A7


2

t

A2

A1A3

A4

A5

A6

A7

Remove all added arcs connected to selected activities§ Re-direct arcs from non-selected to selected activities


2

t

A2

A1A3

A4

A5

A6

A7


We can safely remove those



2

t

A2

A1A3

A4

A5

A6

A7

But we have to re-direct this


This is the relaxed solution (a modified project graph)

A2

A1A3

A4

A5

A6

A7

And we have to complete it to a full schedule

SA-LNS Completion Step

Schedule-or-Postpone cannot be used directly§ because it does not support non-regular objectives (scheduling at

the EST is not a good idea)

Solution: solve a Linear Program (poly-time)§ Relax all resource constraints§ The solution provides a set of target start times

Then run a modified Schedule-or-Postpone§ Start activities at the target start time, instead of the EST§ De-postpone activity when propagation prunes the target start time

Self-Adaptive

Self-Adaptive

Find a solution

Relax a fragment


improved?


no yes

§ Relaxation heuristic and completion strategy are chosen according to a probability distribution

Rel. heuristic selector

Compl. strategy selector

Modify selection probability

§ The distribution is updated based on the obtained improvement

Self-Adaptive

Find a solution

Relax a fragment


improved?


no yes

Rel. heuristic selector

Compl. strategy selector

Modify selection probability

On average 4% from the optimal solution on 20 diverse benchmarks

This is the search strategy employed by the system you will use in the hands-on session

Self-Adaptive§ Relaxation heuristic and completion strategy are chosen according

to a probability distribution§ The distribution is updated based on the obtained improvement



Who is the winner?

Who is the Winner?

We have seen a number of techniques: what’s the best?

It’s problem dependent!

§ Some techniques work well on smaller problems, but do not scale well (e.g. MCS enumeration)

§ Some techniques work well for certain problem features (e.g. Schedule-or-postpone for the classical RCPSP)

§ Some techniques are more robust, but they got beaten on specific problem variants

§ ...

Lesson 1 is the “standard” for Constraint Based Scheduling§ Schedule-or-postpone + timetabling and (extended) edge-finding§ Timetable edge-finding is the current state of the art for edge-

finding like filtering§ Precedence graph based propagation is used in CP Optimizer for

unary resources (and works very well)§ No-first/not-last rules are seldom used. Their current stat-of-the-art

implementation is [4]

That Said...

About this lesson:§ PCP (see work [11]) was the state of the art for the classical RCPSP

until very recently§ Self-adapting Large-Neighborhood Search is the current state of

the art as for robustness

However:§ The state of the art for classical RCPSP is Lazy Clause Generation§ LGC is a hybrid technique, integrating CP and SAT solvers

That Said...

0

25

50

75

100

J30 J60 J90 J120

MCSLCG

% s

olve

d in

stan

ces

However:§ The state of the art for classical RCPSP is Lazy Clause Generation§ LGC is a hybrid technique, integrating CP and SAT solvers

That Said...

§ Expressiveness and global constraints from CP§ no-good learning from SAT§ search: mix of CP and SAT§ quite complex to implement§ If you are interested, check [16,17]


Lesson 2:Advanced Scheduling Topics

References

References

[8] Grimes, D., & Hebrard, E. (2011). Models and strategies for variants of the job shop scheduling problem. Proc. of CP[9] Igelmund, G., & Radermacher, F. J. (1983). Algorithmic approaches to preselective strategies for stochastic scheduling problems. Networks, 13(1), 29–48.[10] Cesta, A., Oddi, A., & Smith, S. F. (1998). Scheduling Multi-Capacitated Resources under Complex Temporal Constraints. Proc. of CP, 465–465.[10bix] Policella, N., Cesta, A., Oddi, A., & Smith, S. F. (2007). From precedence constraint posting to partial order schedules: A CSP approach to Robust Scheduling. AI Communications, 20(3), 163–180.[11] Laborie, P. (2005). Complete MCS-Based Search: Application to Resource Constrained Project Scheduling. Proc. of IJCAI (pp. 181–186). Professional Book Center.

References

[12] Lombardi, M., & Milano, M. (2012). A min-flow algorithm for Minimal Critical Set detection in Resource Constrained Project Scheduling. Artificial Intelligence, 182-183, 58–67

References

[13] Policella, N., Smith, S. F., Cesta, A., & Oddi, A. (2004). Generating Robust Schedules through Temporal Flexibility. Proc. of ICAPS (pp. 209–218).[14] Laborie, P., & Godard, D. (2007). Self-adapting large neighborhood search: Application to single-mode scheduling problems. Proc. of MISTA. [15] Godard, D., Laborie, P., & Nuijten, W. (2005). Randomized Large Neighborhood Search for Cumulative Scheduling. Proc. of ICAPS (pp. 81–89). [16] Schutt, A., Feydy, T., Stuckey, P. J., & Wallace, M. (2009). Why Cumulative Decomposition Is Not as Bad as It Sounds. Proc. of CP (pp. 746–761).[17] Schutt, A., Feydy, T., Stuckey, P., & Wallace, M. (2011). Explaining the cumulative propagator. Constraints.

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