Scheduling and Project Planning

Rolf Möhringrolf.moehring@tu-berlin.de

ADM IIIWS 2015/16

DFG Research Center MATHEON mathematics for key technologies

My expertise in industrial problems

Scheduling in productionand traffic

Routing in traffic, logisticsand telecommunication

Projects in scheduling and logistics

Routing of AGVs in the Hamburg harbor

Ship Traffic Optimization for the Kiel Canal‣

‣

‣ Scheduling and logistics in steel production

P r o d u k t i o n s m a n a g e m e n t M e t a l s

Wir sorgen für ein Wu n d e rw e r k .

‣ Optimizing throughput at a dairy filling line

‣ Turnaround scheduling in chemical plants

Areas needed for solving industrial problems

Mathematics Computer

Science

Engineering&

Economics

Industrial Problems

Theory Applications

Overview

‣ Part I: Sequencing and Scheduling

Example 1: Slab logistics

Example 2: Coil coating

Example 3: Dairy production filling line

‣ Part II: Scheduling under Uncertainty

Approaches to uncertainty

Example 4: Shutdown and Turnaround Scheduling

Sequencing and Scheduling

input of n items

sequencethem

schedule them w.r.t. the sequence

cost depends on both

conditions may depend on subsequences

Example 2: Coil coating [Höhn, König, Lübbecke, Möhring 2008-09, published 2011]

coils need to be sequenced

run through coating line

complex scheduling with shuttle coaters

74 Coil Coating with Shuttles

shaping steel producers’ extremely diverse product portfolio: The coils usedfor home appliances, for instance, already have their typical white coatingwhen bought from the steel supplier; the sheet metal used for car bodiesalready has an anti-corrosion coating before it arrives at the automotive plantfor pressing; coils destined for building construction receive their coatings,which are very specific for technical as well as esthetic reasons, while still atthe steel plant.

Figure 5.1: Some coils of sheet metal. They comprise up to several hundredmeters of sheet metal, and may weigh as much as twenty tons.

Steel producers and manufacturers of coating materials on the one hand,and distributors of pre-coated sheet metal on the other hand, have formedassociations to promote the evolution of coil coating on national [82] andinternational [43] levels already in the 1960s. Progress in the development ofnew and improved coating materials and techniques fosters an ongoing diver-sification in pre-coated metal products, and in recent years there have beenquite a few scientific publications on coil coating, e.g., [39, 80]. Yet, to thebest of our knowledge, the present work is the first dealing with optimizationin the planning process.

As is typical for paint jobs, the coil coating process may be subject tolong setup times, mainly for the cleaning of equipment, and thus very highsetup cost. In order to reduce this cost, so-called shuttle coaters have beenintroduced. They possess two separate tanks which allow to hold two di�erentcoatings at the same time, see Figure 5.2. The advantage is twofold: The

Problem Model Algorithms

Tank Assignment Problem

Subproblem: given fixed-order coil sequence

,find tank assignment with minimum total idle time

2

1

rollerchange

rollerchange & roller

change

cleaning cleaning

cleaning

cleaning

Setup work necessary if

color changes cleaning

coil has larger width than predecessor roller change

� concurrent setup work on idle tank saves idle time

Concurrent Setup Scheduling W. Hohn, F. Konig, M. Lubbecke, R. Mohring

primer coater

oven

Author: Integrated Sequencing and Scheduling in Coil CoatingManagement Science 00(0), pp. 000–000, c� 0000 INFORMS 3

as a black box, our algorithm can be adapted easily to other coil coating lines with di�erent setuprules and a di�erent number of shuttle coaters.

2. Problem FormulationSteel coils are a highly individualized product, and all non-productive time in coil coating dependson certain characteristics of coils, so we first state the most important ones for a better under-standing. Coils usually have a length of 1–5 km, take 20–100 minutes to run through the coatingline, and some of their central attributes are naturally the colors they receive in the four coatingstages, chosen from a palette of several hundred, and their width, usually 1–1.8 m. We refrain fromlisting further coil attributes, since their list is very long; yet all relevant information is includedin our calculations.

For a concise description of the optimization task, we shall briefly familiarize the reader withsome coil coating terminology:

• A coater comprises all machinery necessary for applying one of the four layers of color to thecoils, primer and finish on top and bottom.

• A tank holds the color currently in use at a coater, and each tank has its own rubber rollerfor applying the color to the coil.

• Naturally, a coater has at least one tank. A shuttle coater has two tanks, each with its ownroller, which can be used alternatingly to apply color.

• A color change refers to cleaning a tank and filling it with a new color. When a tank is usedto coat a coil wider than its predecessor on that tank, it needs to be replaced, and this is referredto as a roller change.Before entering production, each coil is unrolled and stapled to the end of its predecessor. Duringall non-productive time, scrap coils are inserted in between actual coils, so essentially a never-ending strip of sheet metal is continuously running through the coil coating line. After undergoingsome chemical conditioning of their surface, the coils run through a top and bottom primer coater,an oven, a top and bottom finish coater, and through a second oven. In the ovens, the respectivecoating layers are fixed. After the coating process, the coils are rolled up again, now ready forshipping. A schematic view of a typical coil coating line is depicted in Fig. 2.

chem coater

oven

oven

slingbu�er

sling bu�er

primer coater

finish coater

Figure 2 Schematic view of a coil coating line with chem, primer, and finish coater. The chem and the bottomfinish coaters are standard coaters, the remaining have shuttles.

An instance of our optimization problem comprises a set [n] := {1, . . . , n} of coils to be coated.For the account in this paper, each coil j is characterized by its colors c(j) �Nm on the m coaters, itswidth wj �R+, and its processing time pj �R+. The optimization goal is to minimize the makespanfor coating the given set of coils, i.e., the completion time of the last coil in the sequence. This isessentially equivalent to minimizing non-productive time, or cost, in the plan, which ensues for tworeasons:

sling buffer

sling buffer

oven

finish coater

chem coater

setups in the scheduling phase

2 color tanks

‣ Setup work necessary ifcolor changes → cleaningcoil has larger width than predecessor → roller change

‣ → concurrent setup work on idle tank saves idle time

‣ Subproblem:given fixed-order coil sequence, find tank assignment with minimum total idle time

Details about the scheduling phaseProblem Model Algorithms

Tank Assignment Problem

Subproblem: given fixed-order coil sequence,find tank assignment with minimum total idle time

tank 2

tank 1

Setup work necessary if

color changes cleaning

coil has larger width than predecessor roller change

� concurrent setup work on idle tank saves idle time

Concurrent Setup Scheduling W. Hohn, F. Konig, M. Lubbecke, R. Mohring

Problem Model Algorithms

Tank Assignment Problem

Subproblem: given fixed-order coil sequence,find tank assignment with minimum total idle time

tank 2

tank 1

Setup work necessary if

color changes cleaning

coil has larger width than predecessor roller change

� concurrent setup work on idle tank saves idle time

Concurrent Setup Scheduling W. Hohn, F. Konig, M. Lubbecke, R. Mohring

Problem Model Algorithms

Tank Assignment Problem

Subproblem: given fixed-order coil sequence,find tank assignment with minimum total idle time

tank 2

tank 1

Setup work necessary if

color changes cleaning

coil has larger width than predecessor roller change

� concurrent setup work on idle tank saves idle time

Concurrent Setup Scheduling W. Hohn, F. Konig, M. Lubbecke, R. Mohring

Problem Model Algorithms

Tank Assignment Problem

Subproblem: given fixed-order coil sequence,find tank assignment with minimum total idle time

tank 2

tank 1

Setup work necessary if

color changes cleaning

coil has larger width than predecessor roller change

� concurrent setup work on idle tank saves idle time

Concurrent Setup Scheduling W. Hohn, F. Konig, M. Lubbecke, R. Mohring

Problem Model Algorithms

Tank Assignment Problem

Subproblem: given fixed-order coil sequence

,find tank assignment with minimum total idle time

2

1

rollerchange

rollerchange & roller

change

cleaning cleaning

cleaning

cleaning

Setup work necessary if

color changes cleaning

coil has larger width than predecessor roller change

� concurrent setup work on idle tank saves idle time

Concurrent Setup Scheduling W. Hohn, F. Konig, M. Lubbecke, R. Mohring

Problem Model Algorithms

Tank Assignment Problem

Subproblem: given fixed-order coil sequence

,find tank assignment with minimum total idle time

2

1

rollerchange

rollerchange & roller

change

cleaning cleaning

cleaning

cleaning

Setup work necessary if

color changes cleaning

coil has larger width than predecessor roller change

� concurrent setup work on idle tank saves idle time

Concurrent Setup Scheduling W. Hohn, F. Konig, M. Lubbecke, R. Mohring

Graph model for the scheduling phase

Problem Model Algorithms

Tank Assignment Problem for k Shuttle Coaters

chem coater

oven

oven

primer coater

finish coater

finishcoater

General problem contains k Shuttle Coaters:

k shuttle coaters

tank 1

tank 2

tank 1

tank 2

tank 1

tank 2

no parallel concurrent setup

Concurrent Setup Scheduling W. Hohn, F. Konig, M. Lubbecke, R. Mohring

Problem Model Algorithms

Overview

Practice

Theory

e�cient algorithmnew ideas for

small instanceseven for

far too slow,

polynomial-time algorithm

dynamic programmingfor fixed k

strongly NP-hardin special 2-union graphsMax Weight Indep. Set

Problem with k coatersTank Assignment

Concurrent Setup Scheduling W. Hohn, F. Konig, M. Lubbecke, R. Mohring

‣ k shuttle coaters

‣ no parallel concurrent setup

Problem Model Algorithms

A Model for Tank Assignment Optimization

Special case: one shuttle coater

� increase interval weights bysavings due to concurrent setup

� optimum tank assignment⇥ max weight independent set⇥ in interval graph

� can be solved in polynomial time

General case: k shuttle coaters

� need generalized intervals due toconcurrent setup

� special class of 2-union graphs� optimum tank assignment⇥ max weight independent set⇥ in special 2-union graph

� NP-hard

machine 2

machine 3

machine 1

tim

e

Concurrent Setup Scheduling W. Hohn, F. Konig, M. Lubbecke, R. Mohring

Practice

Theory

chem coater top

chem coater bottom

primer top, tank 1

primer top, tank 2

primer bottom, tank 2

primer bottom, tank 1

finish top, tank 1

finish top, tank 2

finish bottom

6:00h 10:15h 12:30h 14:45h 17:00h 19:15h 21:30h 23:45h 2:00h 4:15h 6:30h 8:45h 11:00h 13:15h 15:30h 17:45h 20:00h 22:15h

Combining sequencing and scheduling

visualization by Gantt chart, throughput ≈ makespan

Sequence generation with a fast genetic algorithm

Scheduling based on the insights from dyn. prog.

Quality testing by lower bounds

Author: Integrated Sequencing and Scheduling in Coil Coating20 Management Science 00(0), pp. 000–000, c� 0000 INFORMS

0.00

0.20

0.40

0.60

0.80

1.00

1.20

best alphaalpha = 0.5FIFO

Figure 9 Comparison of normalized non-productive time included in our long-term solutions. From left to right,costs are for our algorithm when using the independent set heuristic with optimal choice of �, with fixed choice of

� = 0.5, and when using the FIFO online rule.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

trivial lower boundTSP lower boundIP lower boundFIFOexpert planner

Figure 10 Comparison of normalized bounds and makespans for representative short-term instances whenrestricted to FIFO tank assignment. Bounds displayed from left to right are LBtriv, the sum of coil processing

times, LBTSP, the sum of processing times plus local cost in an optimal, local cost based TSP solution, and ourIP bound LBIP. Makespans are given for our solutions obtained using the FIFO online rule for scheduling, as well

as for a reference solution devised by an expert human planner where it was available.

7.2.2. Short-Term Instances For all short term instances, we succeeded in computing lowerbounds by our branch-and-price approach, proving our solutions to be within at most 10% ofmakespan optimality, cf. Fig. 10. Yet, we did not solve all instances to integer optimality and alsoused short subsequences only (� � 6), so the lower bound is certainly improvable.

If we concentrate on cost—in contrast to makespan—it can be seen from Fig. 11 that the integerprogramming lower bound is able to close much more of the gap to the upper bound than the TSPbound which is optimal w.r.t. local cost only.

Finally, the superiority of the independent set heuristic to FIFO is less significant in short-termplanning. While both heuristics were on a par for most instances, small improvements over FIFOwere observed in three cases.

8. Summary and ConclusionsWe have developed an exact mathematical model for the complex integrated sequencing andscheduling task in coil coating with shuttles and implemented optimization software solving it in

our algorithmexpert planner (human)

Author: Integrated Sequencing and Scheduling in Coil Coating20 Management Science 00(0), pp. 000–000, c� 0000 INFORMS

0.00

0.20

0.40

0.60

0.80

1.00

1.20

best alphaalpha = 0.5FIFO

Figure 9 Comparison of normalized non-productive time included in our long-term solutions. From left to right,costs are for our algorithm when using the independent set heuristic with optimal choice of �, with fixed choice of

� = 0.5, and when using the FIFO online rule.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

trivial lower boundTSP lower boundIP lower boundFIFOexpert planner

Figure 10 Comparison of normalized bounds and makespans for representative short-term instances whenrestricted to FIFO tank assignment. Bounds displayed from left to right are LBtriv, the sum of coil processing

times, LBTSP, the sum of processing times plus local cost in an optimal, local cost based TSP solution, and ourIP bound LBIP. Makespans are given for our solutions obtained using the FIFO online rule for scheduling, as well

as for a reference solution devised by an expert human planner where it was available.

7.2.2. Short-Term Instances For all short term instances, we succeeded in computing lowerbounds by our branch-and-price approach, proving our solutions to be within at most 10% ofmakespan optimality, cf. Fig. 10. Yet, we did not solve all instances to integer optimality and alsoused short subsequences only (� � 6), so the lower bound is certainly improvable.

If we concentrate on cost—in contrast to makespan—it can be seen from Fig. 11 that the integerprogramming lower bound is able to close much more of the gap to the upper bound than the TSPbound which is optimal w.r.t. local cost only.

Finally, the superiority of the independent set heuristic to FIFO is less significant in short-termplanning. While both heuristics were on a par for most instances, small improvements over FIFOwere observed in three cases.

8. Summary and ConclusionsWe have developed an exact mathematical model for the complex integrated sequencing andscheduling task in coil coating with shuttles and implemented optimization software solving it in

Author: Integrated Sequencing and Scheduling in Coil Coating20 Management Science 00(0), pp. 000–000, c� 0000 INFORMS

0.00

0.20

0.40

0.60

0.80

1.00

1.20

best alphaalpha = 0.5FIFO

Figure 9 Comparison of normalized non-productive time included in our long-term solutions. From left to right,costs are for our algorithm when using the independent set heuristic with optimal choice of �, with fixed choice of

� = 0.5, and when using the FIFO online rule.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

trivial lower boundTSP lower boundIP lower boundFIFOexpert planner

Figure 10 Comparison of normalized bounds and makespans for representative short-term instances whenrestricted to FIFO tank assignment. Bounds displayed from left to right are LBtriv, the sum of coil processing

times, LBTSP, the sum of processing times plus local cost in an optimal, local cost based TSP solution, and ourIP bound LBIP. Makespans are given for our solutions obtained using the FIFO online rule for scheduling, as well

as for a reference solution devised by an expert human planner where it was available.

7.2.2. Short-Term Instances For all short term instances, we succeeded in computing lowerbounds by our branch-and-price approach, proving our solutions to be within at most 10% ofmakespan optimality, cf. Fig. 10. Yet, we did not solve all instances to integer optimality and alsoused short subsequences only (� � 6), so the lower bound is certainly improvable.

If we concentrate on cost—in contrast to makespan—it can be seen from Fig. 11 that the integerprogramming lower bound is able to close much more of the gap to the upper bound than the TSPbound which is optimal w.r.t. local cost only.

Finally, the superiority of the independent set heuristic to FIFO is less significant in short-termplanning. While both heuristics were on a par for most instances, small improvements over FIFOwere observed in three cases.

8. Summary and ConclusionsWe have developed an exact mathematical model for the complex integrated sequencing andscheduling task in coil coating with shuttles and implemented optimization software solving it in

trivial lower boundTSP lower boundIP lower bound

More expensive and longer ...

‣ Opening delayed since June 2012

‣ Expected to be (much) more expensive

Airport Berlin-Brandenburg

‣ Not ready for the move from Bonn to Berlin

‣ Expected to be (much) more expensive

chancellor’s office by € 50,000,000

More expensive and longer ...

Government & parliament buildings in Berlin, July 1999

‣ Planning and optimization assumes certainty about project details and data

deterministic models prevail

‣ Real life is subject to many influences that are beyond control

machine breakdowns, weather, illness, data deficiencies, …

‣ leads to unreliable results and plans

underestimation of expected makespan and cost

‣ Need models and techniques to cope with uncertainty

Reasons

‣ Online optimization

competitive analysis, smoothed analysis

What could I have achieved had I known the future?

‣ Stochastic dynamic optimization

planning with policies, minimizing in expectation

What can I achieve without knowing the future?

‣ Stochastic programming

limits policies to recourse, uncertainty is exogenous

Partially decide now, have corrective actions later

Models for dealing with uncertainty (1) The underestimation error

Consider only precedence constraints

Fulkerson 1962

Job j has random duration Xj with mean xj ⇒ Cmax(x1, . . . ,xn) ≤ E[Cmax(X1, . . . ,Xn)]

1

2

n

Xj independent,

uniformly distributedon [0,2]

0 1 2

1

Cmax(x1, . . . ,xn) = 1

Q{X j ≤ t} =t2

Q{Cmax ≤ t} =! t

2

"n

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

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

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 exceeding t

‣ Phase 2: calculate a schedule S for tresource leveling

risk analysis of S unforeseen repairs may occur

Features of turnaround projects

‣ They are largea single cracker has 500 – 2000 jobsa turnaround has up to 150,000 jobs

‣ Their precedence constraints are simplelargely series-parallel with some time-lag constraints

‣ Resources are mostly workersplant workers are available at no extra costmore can be hired externally

‣ Risk is imminent inspection may trigger repair work

high out of service cost when length t is exceeded

An example: turnaround of a cracker (1)

‣ ~2000 jobs, turnaround length 4 – 8 days

very detailed, large variation in processing time

must respect shifts

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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.

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KOLONNE SPRITZEN KOL. SPR 6,48 Stunden 1.H.

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An example: turnaround of a cracker (2)

‣ 14 resource types, only few need to be leveled

Pipe fitters (Rohrschlosser), high pressure cleaning

‣ structure of precedence constraints

mostly series parallel with short chains

m = 2

A model for scheduling problems

1

2

3

4

5

6

7

G

12

34

5 67

‣ 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 cost”

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.

Stochastic dynamic optimization

‣ fixed number of jobs

‣ know joint distribution Q of processing times

‣ plan with non-anticipative policies Π

‣ may exploit history and a priory knowledge about Q

‣ Find policy that minimizes / approximates expected performance

performance ratio of policy Π is

inf{E[��] | � policy }

E[��]E[��OPT ]

Comparison with online analysis

1

2

3

3 vectors of processing times, each with prob. 1/3

(1,1,2) (1,2,1) (2,1,1)

minimize Cmax

3 possible schedules when we start with 1 and 2

312

1 32

12 3

Optimal in expectation, and so ratio is 1Competitive (average) analysis gives ratios ( ) > 13

276

Offline optimum is not a policy

E[C�max] = 7

3

1 3

2 4

m = 2 machines

, independent

mimimize E[∑penalties]

Xj � exp(�)

‣ common due date d

‣ penalties for lateness: v for job 2, w for jobs 3,4, v << w

‣ no interruption of jobs

Policies: an example

Start jobs 1 and 2 at t = 0

Danger: job 2 blocks machine

1 3

2 4

12 2

d0

expensive jobs 3 and 4 only sequentially

Example continued

1 3

2 4

d0

Start only job 1 and wait for its completion

Danger: deadline is approaching

1

can do expensive jobs 3 and 4 in parallel

short span to deadline

Example continued

1 3

2 4

d0

start 1 at 0, fix tentative decision time t panic

if C1 ≤ t panic start 3 and 4, and then 2

t panic

1 34 2

d0 t panic

else start 2 at t panic and then 3 and 4

1

2 43

Example continued

0.5 1.5 2.51 2 3

43.2

43.4

42.8

42.6

Expected cost for λ = 1, d = 3, v = 10, w = 100

Example completed

t panic

t panic = 0 and ∞ gives the other two policies

Modeling uncertainty for the shutdown project

‣ Jobs have discrete distributions with 4 processing times

early completion : 5%

normal time : 70 %

surplus for repair : 20%

additional surplus for getting spare parts : 5%

‣ Percentages may change per group of jobs

uses experience from previous turnarounds

and age of equipment

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 bounds for the „rough“ schedules

let the manager decide

manager can change t and see the risk change

‣ Computation uses the stochastic risk measures on the makespan

t

cost and risk

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