Stochastic SchedulingHistory and Challenges
Rolf Möhring
Eurandom Workshop on Scheduling under Uncertainty
Eindhoven, 3 June 2015
DFG Research Center MATHEON mathematics for key technologies
Overview
‣ Part I: Policies for Stochastic Scheduling
The Model
Classes of Policies
Optimality and Approximation Results for Policies
‣ Part II: Risk Analysis for a Policy
Analyzing the Makespan Distribution
Bounds by Network Modification
Dealing with Stochastic Dependencies
‣ Turnaround Scheduling as example
Shutdown and Turnaround Scheduling
Turnaround Scheduling: resource allocation and scheduling of large-scale maintenance activities in chemical manufacturing
Turnaround Scheduling
Turnaround Scheduling: resource allocation and scheduling oflarge-scale maintenance activities in chemical manufacturing
◃ Flexible resource usage! time-cost tradeoff problem
◃ To determine! optimal project duration (cost for
resource usage vs. out-of-service cost)! optimal resource usage (capacity bounds,
resource levelling)
◃ Industrial cooperation2006: developed combinatorial algorithm
2007: add. constraints, stochastic analysis
! Project within DFG Priority Program: Algorithm Engineering.
B13 – Optimization under uncertainty in logistics and scheduling 5 / 13
‣ Phase 1: plan the schedule length tcan hire external workers balance turnaround cost vs. out of service cost for testimate the risk of meeting t
‣ Phase 2: calculate a schedule S for tresource leveling
risk analysis of S unforeseen repairs may occur
with Nicole Megow & Jens Schulz
Ohne Titel
AUERLUEFTER DEMONT LUEF.DEM 2.R.
01.STOSS DEM.Boden 06-01 01. DEM 7,6 Stunden 2.R.
02.STOSS DEM.Boden 14-07 02. DEM 1,79 Tage 2.R.
KOL.TEILE ABLASSEN KT. ABL 7,6 Stunden 2.R.
KOL.-TEILE TRANSP KT. TRA 1,3Std. F
KOL.-TEILE SPRITZEN KT. SPR 41 Minuten 1.H.
KOL.-TEILE KONT.+REP KT.KONTR 2.R.
KOLONNE SPRITZEN KOL. SPR 6,48 Stunden 1.H.
IU EIGENUEBERWACHUNG ABNAH.IU S
KOL.-TEILE AUFZIEHEN KT. AUF 2 Tage 2.R.
01.STOSS MON.Boden 01-06 01. MON 2 Tage 2.R.
02.STOSS MON.Boden 07-14 02 MON 2 Tage 2.R.
MANNLOCHDECKEL MONT ML. MON 2.R.
DRUCKPROBE MIT EDELWASSER DP M.EDELW 0Std. 2.R.
12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16Sa., 24.04. So., 25.04.10 Mo., 26.04.10 Di., 27.04.10 Mi., 28.04.10 Do., 29.04.10 Fr., 30.04.10 Sa., 01.05.10 So., 02.05.10
An example: turnaround of a cracker (1)
‣ ~2000 jobs, turnaround length 4 – 8 days
very detailed, large variation in processing time
must respect workers’ shifts
Overview
‣ Part I: Policies for Stochastic Scheduling
The Model
Classes of Policies
Optimality and Approximation Results for Policies
‣ Part II: Risk Analysis for a Policy
Analyzing the Makespan Distribution
Bounds by Network Modification
Dealing with Stochastic Dependencies
‣ Turnaround Scheduling as example
☞
m = 2
Scheduling problems
1
2
3
4
5
6
7
G
12
345 6
7
‣ a set V of jobs j = 1,…,n (with or without preemption)
‣ a graph (partial order) G of precedence constraints
‣ resource constraints, here given by m identical machines
‣ release dates rj
Model
Uncertainty and objective
‣ jobs have random processing times Xj with known
distribution Qj
we know their joint distribution
‣ “cost” function κ( C1,…,Cn ) depending on the (random)
completion times C1,…,Cn
examples Cmax , ∑ wj Cj , ∑ wj Fj ( Fj = Cj – rj )
‣ plan jobs over time and “minimize expected cost”
“What can I achieve without knowing the future”
Planning with policies – the dynamic view
time
Decision at time t (non-anticipative)
t
S(t)
start set S(t) (possibly empty)
fix tentative next decision time t plan. (deliberate idleness)
next decision time = min { t plan., next completion time }
t plan.
Data deficiencies, use of approximate methods (simulation)require stability condition:
Stability of policies
No stability for optimal policies in general!
Q‘ approximates Qκ‘ approximates κ
OPT(Q‘,κ ‘) approximates OPT(Q,κ)
xε
1
1
y
1 13
2
34
52
43 5
1
2
3
4
5
Excessive use of information yields instability
min E(Cmax)
Q� :�
x� = (1 + �, 4, 4, 8, 4)y = (1, 4, 4, 4, 8)
with prob. ½with prob. ½
EQ�(Cmax) = 13
for ε → 0
1
2
3
4
5Q :
x = (1, 4, 4,8,4) with probability 12
y = (1,4, 4,4,8) with probability 12
! " #
$ #
1
12
253
4
43 5
ε → 0 ⇒ Qε → Q with
No info when 1 completes. So start 2 at t = 0
y
x
12 16
EQ(Cmax)⇥= 13= lim��0 EQ�(Cmax)
Overview uncertainty in scheduling
‣ Part I: Policies for Stochastic Scheduling
The Model
Classes of Policies
Optimality and Approximation Results for Policies
‣ Part II: Risk Analysis for a Policy
Analyzing the Makespan Distribution
Bounds by Network Modification
Dealing with Stochastic Dependencies
‣ Turnaround Scheduling as example
☞
Therefore: Stable classes of policies
‣ Stability is more important than optimalityuse „good“ robust policies instead
‣ Investigate classes of policies w.r.t. stability and approximation behavior
‣ priority policies (list scheduling) no stability in generalstochastic approximation results available
‣ preselective policies (delay policies)have stabilityonly few results on approximation
3
7
Priority policies have anomalies
1 5
3
4
7
6
2
minimize makespan on 2 identical machines
use priority list 1 < 2 < 3 ...
x = (4,2,2,5,5,10,10)
21
y = x – 1 = (3,1,1,4,4,9,9)
45
67
34 5
61
2
Preselective policies
F is forbidden set :⇔ F cannot be scheduled simultaneously but every proper subset can
Solve conflict on every F by selecting a waiting job jF ∈ F jF must wait until any job from F is completed
i
j
k
Fj selected
i
j
k
OR conditionrepresentinglocal priority
do early start scheduling w.r.t. original precedence constraints and OR conditions resulting from preselected waiting jobs
No anomalies for preselective policies
1 5
3
4
7
6
2
2 identical parallel machines⇒ F = {4,5,7} is only forbidden set
x = (4,2,2,5,5,10,10)
y = x – 1 = (3,1,1,4,4,9,9)
213
45
67
345
12
67
7
Three views on policies
online planning rule
combinatorialobject
make decisions over time
maps processing times to start times
used for computations(special policies only)
function fromRn → Rn
Preselective policies and AND/OR networks
1
2
3
4
5
6
7
G
F : {3,4,5}, {1,4}
F1
F2
4 4
1
23
4
5
6
7F1
F2
AND/OR network representing Π
this choice ofwaiting jobs defines policy Π
start of a job in policy Π = min/max of path lengths= min/max of sums of processing times
⇒ Π is continuous and monotone
F
Tasks related to AND/OR networks
12
4
3
7 8 10F4 F5
F26
5
9F3
F1 ‣ test feasibility of waiting job choice
‣ compute earliest start
‣ detect forced OR conditions (transitivity)
may contain cycles
fast algorithms available [M., Skutella & Stork 2004]
in NP ∩ coNP for arbitrary arc weights, no fast algo known
A special case: partial order policies
i
j
k
Fj selected
i
j
k
OR conditionrepresentinglocal priority
select (i,j)add a precedence constraint on F⇒ only AND conditions
start of a job in policy Π = earliest start w.r.t. to a partial order of precedence constraints= max of sums of processing times
⇒ Π is continuous, monotone, and convex
Nice policies are preselective
‣ Π is monotone iff Π is preselective (up to domination)
‣ Π is continuous iff Π is preselective (up to domination)
‣ Π is convex iff Π is a partial order policy
combinatorialobject
function fromRn → Rn
[Igelmund & Radermacher 1985]
Overview
‣ Part I: Policies for Stochastic Scheduling
The Model
Classes of Policies
Optimality and Approximation Results for Policies
‣ Part II: Risk Analysis for a Policy
Analyzing the Makespan Distribution
Bounds by Network Modification
Dealing with Stochastic Dependencies
‣ Turnaround Scheduling as running example
☞
Some special optimality results
m machines
independent exponentially distributed processing times
– LEPT is optimal for Cmax [Weiss 1982]
– SEPT is optimal for ∑ Cj [Weiss & Pinedo 1982]
– no optimal policy known for ∑ wj Cj
Minimize E(∑ wj Cj) on m identical machines for random
independent processing times
1 machine
WSEPT is optimal for ∑ wj Cj [Rothkopf 1966]
A more general optimality result
‣ Set policy: use only completion, being busy and time t for decisionsno tentative decision times
‣ Additive cost function κ( C1,…,Cn )is the integral over cost rate g(U(t)) of the set U(t) of uncompleted jobs at time t
‣ Xj independent and exponentially distributed⇒ ∃ optimal set policy [M., Radermacher, Weiss 1985]
tplanned
S(t)
t
g( )g( )g( )
The non-idleness problem
‣ Set policies
contain preselective and priority policies
may leave resources idle for obtaining information, even for exponential distributions and m parallel machines
‣ Open challenge
is there an optimal set policy without idle times for ∑ wj Cj on m parallel machines and independent
exponentially distributed Xj
The LP-based approach for
Find a polyhedral relaxation P
Solve the linear program (LP)
min{∑ j w jCLPj | (CLP
1 , . . . ,CLPn ) ∈ P}
(CLP1 , . . . ,CLP
n )i1 < i2 < .. . < inUse the list L:
induced by
for defining a preselective policy
CLPi1 ≤CLP
i2 ≤ . . . ≤CLPin
Consider the achievable region {(E[CΠ1 ], . . . ,E[CΠn ]) ∈ R
n |Π policy}
[M., Schulz & Uetz 1999]
� j w jCj
The LP relaxation
CLPj ≥ E[Xj] for all j ∈V
∑j∈A
E[Xj]CLPj ≥12m
!!
∑j∈A
E[Xj]"2
+ ∑j∈A
E[Xj]2"
−(m−1)(∆−1)
2m
!
∑j∈A
E[Xj]2"
for all A⊆V
Assume CV[Xj]2 ≤ ∆ CV[Xj]2 :=VAR[Xj]E[Xj]2
LP can be solved combinatorially in polynomial time
Results for parallel machine scheduling
Analysis of WSEPT [M., Schulz, Uetz](weighted shortest expected processing time first)
– approximation, combinatorial algorithm, LP approach is crucial
!
2− 1m"
– approximation, LP approach is crucial
Adding release dates [M., Schulz, Uetz]
!
4− 1m"
use job-based priority policy
– approximation, LP approach is crucial
Adding precedence constraints [Skutella, Uetz]
use delayed list scheduling!
3+2β +m−1mβ
"
Extended results
‣ Stochastic online scheduling
jobs arrive online and must be assigned to machines now “next day”, jobs are scheduled on the assigned machines
number of jobs is not known in advance
jobs have random processing times on the machines
better or matching bounds as in previous model for ∑ wj Cj [Megow, Uetz, Vredeveld 06]
‣ Preemptive scheduling on parallel machines
2-approximation for ∑ wj Cj [Megow, Vredeveld 07]
involved analysis, uses Gittins index
‣ [Skutella, Sviridenko, Uetz 2014] (in the context of unrelated machines) The performance ratio of any fixed-assignment policy can be as large as (1−δ)∆ for any δ > 0, for large enough m
‣ [Labonté, M., Megow 2014] for every k, there is an instance Ik with ∆ ≤ k, such that the performance ratio of WSEPT is as large as , for large enough n
Role of the coefficient of variation
Assumed CV[Xj]2 ≤ ∆ ∆ enters performance guarantee
Is this avoidable?
Not for some classes of policies
⌦(4pk)
More on the coefficient of variation
‣ The approximation ratio of SEPT for ∑Cj is Ω(n1/4) and Ω(m)
‣ A clever priority policy achieves O(log2n + m log n) for an arbitrary number of machines
‣ large lower bound implies that n, m, and ∆ grow simultaneously
‣ Challenge:
does SEPT achieve a constant approximation ration for a constant number of machines ?
[Im, Moseley, Pruhs 2015]
Open challenges
‣ Can one beat ∆ ?
i.e. are there policies whose approximation ratio does not depend on ∆ ?
‣ m || ∑ wj Cj with independent exponentially distributed
processing times
Is there an optimal policy without idle times ? Maybe even a priority policy ?
Overview
‣ Part I: Policies for Stochastic Scheduling
The Model
Classes of Policies
Optimality and Approximation Results for Policies
‣ Part II: Risk Analysis for a Policy
Analyzing the Makespan Distribution
Bounds by Network Modification
Dealing with Stochastic Dependencies
‣ Turnaround Scheduling as running example
☞
Detailed analysis of makespan distribution
Ideal: distribution function F of Cmax
Modest: percentiles
makespan
1
0
F(t)
t90
90%
t90
= inf{t | Pr{Cmax
� t} � 0.90}
Obtaining stochastic information is hard
Hagstrom ’88: The problems below are #P-complete
Given: Stochastic project network with discrete independent processing times
Wanted: Expected makespan
MEAN
Given: Stochastic project network with discrete independent processing times, time t
Wanted: Pr{ makespan ≤ t }
DF
Simulation
‣ requires (complete) information about distributions
‣ difficult to model stochastic dependencies
Approximate methods and bounds
Bounding the distribution function
‣ possible with incomplete information
‣ permits stochastic dependencies
‣ combinatorial algorithms
Overview
‣ Part I: Policies for Stochastic Scheduling
The Model
Classes of Policies
Optimality and Approximation Results for Policies
‣ Part II: Analyzing a Policy
Analyzing the Makespan Distribution
Bounds by Network Modification
Dealing with Stochastic Dependencies
‣ Turnaround Scheduling as running example
☞
Arc diagrams
Node diagramjobs are nodes of digraph G
Arc diagramjobs are arcs of digraph Ddummy arcs may be necessary
makespan = longest pathstandard graph algorithms apply
1
4
6
7
5
3
2
DG1
2
3
4
5
6
7
Chain minors
‣ every chain of N is contained in a chain of N’by taking an appropriate copy
N N’
Let N, N’ be project networks.
N is a chain minor of N’ if
‣ every job of N is represented by (one or several) copies in N’
Chain minors: an example
14
7
85
3
2
N
6
N’1
47
85
32
6
1 7
14
7
8
3
5
1 5
1 73 5 6
8
252 6 7
extreme case:parallel composition of all maximal chains
Bounds based on chain minors
‣ Let N be a chain minor of N’
Give copies of a job from N in N’ the same processing time distribution as their original
Take all processing time distributions in N’ as stochastically independent
‣ Then the makespan of N is stochastically smaller than the makespan of N’QC
max
(N) �st QCmax
(N�)
M. & Müller `99
Sandwiching a network with minors
Given N, look for special networks N1, N2, such that
N1 ⊆minor N ⊆minor N2
1
0
F1 F F2
special = easier to calculate the makespan distribution
Series-parallel networks
A network N is series-parallel if it can be reduced by a finite sequence of series and parallel reductions to one job.
series reduction
indegree = outdegree = 1
parallel reduction
Special cases of the chain minor principle
Kleindorfer ‘71
Shogan ‘77
Spelde ‘76
Dodin ‘85
‣ Work for stochastically independent processing times
‣ Have same underlying combinatorial principle
The bounds of Spelde ‘76
‣ uses special series-parallel networks (parallel chains)
‣ yields upper and lower bounds
3 6
1 4 7
85
3
2
N
6
85
2
14
7
14
7
8
3
5
1 5
1 73 5 6
8
252 6 7
disjoint chains⇒ lower bound
all chains⇒ upper bound
Distribution-free evaluation of Spelde’s bounds
Large networks ⇒
‣ chain length ≈ normally distributed
⇒ μ = ∑ μj and σ2 = ∑ σj2 along a chain
‣ ⇒ per job only μj and σj2 required
Problem: #chains may be exponential
Remedie: Consider only relevant chains
Return the product of the k–1 normal distribution functions F
Relevant chains
µmaxµ2-maxµk -max
Apply k-longest path algorithms to determine k
Excellent and fast bounds in practice
Special case: PERT, considers only Y1
Yi = length of i-longest chain w.r.t. mean processing times μj
Pr(Yk � Y1) � �
Yi
Overview
‣ Part I: Policies for Stochastic Scheduling
The Model
Classes of Policies
Optimality and Approximation Results for Policies
‣ Part II: Analyzing a Policy
Analyzing the Makespan Distribution
Bounds by Network Modification
Dealing with Stochastic Dependencies
‣ Turnaround Scheduling as running example
☞
Worst case approach for stochastic dependencies
[Meilijson & Nadas ‘79, Klein-Haneveld ’86]
Consider expected tardiness
of makespan Cmax
ranges over all joint distributions with the given job processing time distributions as marginals
EQ[(Cmax
� t)+] = EQ[max{0, Cmax
� t}]
�(t) = supQ EQ[(Cmax
� t)+]
in the worst case, i.e.
Properties of expected tardiness
X1 2 3
.25
.5
is piecewise linear and convex for discrete random variables X
EQ[(X� t)+]
1 2 3
1
2
t
slope –1
slope –3/4
slope –1/4
EQ[(X� t)+]
EQ[X]
Properties of
special convex separable optimization problem
t0
slope –1 to left of t0
convex, decreasingto right of t0
t
ψ(t)
�(t) = supQ EQ[(Cmax
� t)+]
�(t) = min(x1,...,xn
) �j
E[(X
j
� x
j
)+]
such that Cmax(x1,...,xn) ≤ t
solvable by max flow algorithms for discrete Xj
Overview
‣ Part I: Policies for Stochastic Scheduling
The Model
Classes of Policies
Optimality and Approximation Results for Policies
‣ Part II: Analyzing a Policy
Analyzing the Makespan Distribution
Bounds by Network Modification
Dealing with Stochastic Dependencies
‣ Turnaround Scheduling as running example☞
Solving the turnaround problem: Phase 1
‣ Phase 1: plan the schedule length tsolve a time-cost tradeoff problem, relax shifts and assume continuous workers
use the breakpoints on the time-cost tradeoff curve to calculate feasible schedules heuristically (no resource leveling)yields alternative „rough“ schedules
Constraints
rj dj
processing times
5 / 1
Constraints
rj dj
processing times
5 / 1
cost
time
155,000
150,000
25015050
145,000
„rough“ schedules
Uncertainty in Phase 1
‣ Phase 1: plan the schedule length t
present the risk for the „rough“ schedules
let the manager decide
manager can change t and see the risk change
‣ Computation uses the stochastic bounds on the makespan
t
Solving the turnaround problem: Phase 2
‣ Phase 2: calculates a schedule S for the chosen time t
settles neglected side constraints such as time lags
levels resources heuristically
uses resource flow for defining a network N
calculates the risk based on N
‣ In addition
have compared our algorithm on small instances with results from a MIP solver for a MIP formulation
Result of the leveling algorithm
52%
%
9% 63%
%
33%
%
66%
%
9%
58%
5%
25%
%
12%
%
10% 33% 28% 14% 17% 42% 12% 21% 50% 10% 8% 12% 19% 6% 12% 15% 35% 12% 27% 8% 38% 6% 12% 19% 34% 16% 12% 8%
14% 28% 25% 29% 10% 21% 25% 9% 9% 33% 30% 9% 9% 31% 38% 29% 40% 38%
48% 100
%
174
%
152
%
100
%
26% 38% 100
%
100
%
12%
260
%
394
%
165
%
20% 92% 165
%
126
%
80% 195
%
174
%
26% 390
%
212
%
65% 376
%
99% 80% 108
%
120
%
90% 20% 40% 269
%
372
%
324
%
168
%
339
%
80% 354
%
559
%
189
%
174
%
189
%
304
%
56% 35% 269
%
109
%
152
%
205
%
38% 40% 18% 20% 16% 24%
10% 100
%
100
%
147
%
310
%
20% 10% 2% 6% 8% 8% 8%
48% 72% 42% 19% 25% 25% 46% 8% 40% 27% 21% 25% 26% 15% 44% 8% 31% 18% 51% 25% 95% 82% 25% 10% 78% 33% 51% 64%
22% 10% 2% 2% 2% 8% 2% 1% 1% 5% 2% 2% 2% 5% 2% 2% 2% 2% 2% 2%
160
%
200
%
152
%
37%
83% 216
%
84% 60% 160
%
165
%
62% 61% 99% 48%
3% 78% 36% 25% 12% 25% 25% 12% 12% 12% 12% 12%
15% 23% 25% 12% 29% 12% 12% 17% 38% 12% 38% 12% 12% 38% 30% 32% 12% 38% 16% 9% 12% 42% 8% 18% 20% 12% 25%
0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20
Di, 04.09.07 Mi, 05.09.07 Do, 06.09.07 Fr, 07.09.07 Sa, 08.09.07 So, 09.09.07 Mo, 10.09.07 Di, 11.09.07
144%
229% 185% 125% 93%
25% 90%
%
300
%
33%
%
60%
%
9% 20%
98%
0%
98%
0%
66%
8%
4%
10% 17% 53% 42% 12% 32% 17% 19% 26% 65% 29% 24% 38%
14% 8% 3% 30% 19% 29% 8% 25% 38% 33% 8% 35% 23% 17%
25% 100
%
100
%
75% 100
%
100
%
50% 72% 100
%
100
%
28%
275
%
289
%
25% 174
%
60% 152
%
81% 26% 142
%
9% 131
%
192
%
242
%
299
%
300
%
279
%
268
%
182
%
89% 206
%
140
%
262
%
129
%
254
%
149
%
60% 40% 135
%
251
%
270
%
278
%
262
%
234
%
170
%
289
%
299
%
216
%
225
%
244
%
114
%
11% 26% 38% 4% 35% 40%
38% 100
%
298
%
240
%
8% 8% 10% 8% 14% 5%
99% 40% 33% 19% 48% 25% 8% 11% 26% 45% 113
%
15% 29% 4% 35% 8% 8% 40% 19% 17% 12% 25% 18% 49% 8% 31% 44% 46% 12% 50% 81% 33%
40% 2% 38% 2%
82% 100
%
100
%
100
%
100
%
68%
84% 100
%
100
%
100
%
41% 69% 100
%
100
%
100
%
100
%
62% 83%
9% 3% 12% 12% 12% 12% 68% 27% 42% 12% 54%
4% 25% 17% 12% 38% 12% 38% 55% 70% 12% 25% 12% 40% 48% 38% 12% 50% 12% 25% 18% 7%
0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20
Di, 04.09.07 Mi, 05.09.07 Do, 06.09.07 Fr, 07.09.07 Sa, 08.09.07 So, 09.09.07 Mo, 10.09.07 Di, 11.09.07
100%
100% 100% 100%
unleveled
leveled
Summary
‣ Uncertainty is imminent in practical scheduling problemsthere are good tools available to analyze risks and implement policies
‣ Turnaround problems are an excellent field for many aspects of scheduling
time-cost tradeoff, malleable jobs, multi-moderesource leveling, calendars
‣ Paper in INFORMS J. Computing 2011 (with Nicole Megow and Jens Schulz)
‣ Turnaround instance available ftp://ftp.math.tu-berlin.de/pub/combi/projects/turnaround/
Info:
www.coga.tu-berlin.de
Thanks!
