Table Of ContentLinköping studies in science and technology
Dissertations, No. 1107
Multi-year maintenance optimisation
for paved public roads – segment based modelling
and price-directive decomposition
Per-Åke Andersson
Department of Mathematics
Linköping 2007
Multi-year maintenance optimisation for paved public roads –
– segment based modelling and price-directive decomposition
© 2007 Per-Åke Andersson
Matematiska institutionen
Linköpings universitet
SE-581 83 Linköping, Sweden
peand@mai.liu.se
Linköping studies in science and technology. Dissertations, No. 1107
ISBN: 978-91-85831-73-9
ISSN 0345-7524
Printed by LiU-Tryck, Linköping 2007
Acknowledgements
This project has been administered by CDU, KTH (Centre for research and education in
operation and maintenance of infrastructure, Royal institute of technology) and performed at
MAI, LiU (Department of mathematics, Linköping university), with financial support from VV
(Swedish road administration), Vinnova, VTI (Swedish national road and transport research
institute), Örebro university and my wife Lena.
I thank
• professor em. Sven Erlander, MAI, for giving me a 1st chance and for introducing me
into optimisation and traffic, professor em. Gunnar Aronsson, MAI, for a 2nd and my
main supervisor professor em. Per Olov Lindberg for a 3rd chance and for his patience
and the constructive-minded reading,
• my other supervisors Torbjörn Larsson, MAI, for his wit at meetings, and Lars-Göran
Wågberg, VTI,
• the other members of the project consultative group Hans Cedermark / Håkan
Westerlund, CDU, for their commitment, Jaro Potucek / Arne Johansson, VV – and
Carl-Gösta Enocksson and Johan Lang, VV, both also crucial members of the
modelling and validation group,
• Jörgen Blomvall, MAI, for making my parallelisation runs on the Penta PC-cluster
possible,
• NSC for granting me 48 000 CPU-hours in all on the Monolith PC-cluster,
• all personnel at MAI, especially the optimisation group, for their kindness and service-
mindedness, and Tommy Elfving, MAI, for his encouragement.
Linköping May, 2007
Per-Åke Andersson
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Abstract
The thesis deals with the generation of cost efficient maintenance plans for
paved roads, based on database information about the current surface conditions
and functional models for costs and state changes, partly developed in co-
operation with Vägverket (VV, Swedish Road Administration). The intended
use is in a stage of budgeting and planning, before concrete project information
is available. Unlike the up to now used models, individual maintenance plans
can be formulated for each segment (a homogeneous road section as to the cur-
rent pavement state and paving history), in continuous state and works spaces.
By using Lagrangean relaxation optimisation techniques, the special benefit/-
cost-ratio constraints that VV puts on each maintenance project can be naturally
mastered by dual prices for the budget constraints. The number of segments
competing for budget resources is usually large. Data from VV Vägdatabank
(SRA Road Database) in county Värmland were used, comprising around 9000
road segments. Due to the large data amount the implemented programs rely on
parallel computation. During the thesis work, access to the PC-cluster Monolith
at NSC was granted. In order to reduce optimisation run times, model & method
development was needed. By aggregating the road segments into road classes,
good initial values of the dual prices were achieved. By adding new state
dimensions, the use of the Markov property could be motivated. By developing
a special residual value routine, the explicitly considered time period could be
reduced. At solving the dual subproblem special attention was paid to the
discretization effects in the dynamic programming approach. One type of study
is on a sub-network, e.g. a road. Validation studies were performed on road 63
in Värmland – with promising but not satisfactory results (see below). A special
model for co-ordinated maintenance considers the fine-tuned cost effects of
simultaneous maintenance of contiguous road segments. The other main type of
study is for a whole network. Several method types have been applied, both for
solving the relaxed optimisation problems and for generating maintenance plans
that fit to the budgets. For a decent discretization, the run time for the whole
Värmland network is less than 80 CPU-hrs.A posterior primal heuristics reduces
the demands for parallel processing to a small PC-cluster.The thesis further
studies the effects of redistributing budget means, as well as turning to a trans-
parent stochastic model – both showing modest deviations from the basic model.
Optimisation results for Värmland indicate budget levels around 40% of the
actual Värmland budget as sufficient. However, important cost triggers are
missing in this first model round, e.g., certain functional performance (safety),
all environmental performance (noise etc.) and structural performance (e.g.
bearing capacity, only modelled by an age measure). For increased credibility of
PMS in general and optimisation in particular, the discrepancies should be
further analysed and lead to improvements as to condition monitoring, state
effect & cost modelling and mathematical modelling & implementation.
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Sammanfattning
I avhandlingen studeras hur kostnadseffektiva underhålls- (uh-)planer för belagd
väg kan genereras, på basis av information om aktuellt vägytetillstånd och
funktionella modeller för kostnads- och tillståndsförändringar, delvis utvecklade
i samarbete med svenska Vägverket (VV). Tilltänkt användning är på strategisk
och programnivå, innan mer detaljerad objektinformation finns att tillgå. Till
skillnad från hittills använda modeller, så genereras individuella uh-planer för
varje vägsegment (en homogen vägsträcka vad gäller aktuellt beläggnings-
tillstånd och beläggningshistorik), i kontinuerliga tillstånds- och åtgärdsrum.
Genom användning av Lagrangerelaxerande optimeringsteknik, så kan de
speciella nytto/kostnads-kvot-villkor som VV ålägger varje uh-objekt naturligen
hanteras med dualpriser för budgetvillkoren. Antalet vägsegment som
konkurrerar om budgetmedlen är vanligtvis stort. Data från VV:s Vägdatabank
för Värmland har använts, omfattande ca 9000 vägsegment. Genom den stora
datamängden har datorprogrammen implementerats för parallellbearbetning.
Under avhandlingsarbetet har projektet beviljats tillgång till Monolith PC-
klustret vid NSC. För att kunna reducera optimeringskörtiderna har modell- och
metodutveckling varit nödvändig. Genom att aggregera vägsegmenten till
vägklasser har goda startvärden på dualpriserna erhållits. Genom utvecklingen
av en speciell restvärdesrutin har den explicit behandlade tidsperioden kunnat
reduceras. Vid lösandet av det duala subproblemet har speciell uppmärksamhet
ägnats åt de diskretiseringseffekter som uppstår i metoden dynamisk program-
mering. En typ av tillämpning avser ett delvägnät, exempelvis en väg. Valid-
eringsstudier har genomförts på väg 63 i Värmland – med lovande men inte
tillfredsställande resultat (se nedan). En speciell modell för samordnat uh
beaktar stordriftsfördelarna vid samtidig åtgärd på en hel vägsträcka. Den andra
huvudtypen av studier gäller ett helt nätverk. Flera metodtyper har tillämpats,
både för att lösa de relaxerade optimeringsproblemen och för att generera uh-
planer som uppfyller budgetvillkoren. För en anständig diskretisering är
körtiderna för hela Värmland mindre än 80 CPU-timmar. Genom en a posteriori
primal heuristik reduceras kraven på parallellbearbetning till ett litet PC-kluster.
Avhandlingen studerar vidare effekterna av omfördelade budgetmedel samt en
övergång till en transparent, stokastisk modell – vilka båda visar små avvikelser
från basmodellen.
Optimeringsresultaten för Värmland indikerar att budgetnivåer på ca 40% av
Värmlands verkliga uh-budget är tillräckliga. Dock saknas viktiga kostnads-
drivande faktorer i denna första modellomgång, exempelvis vissa funktionella
prestanda (säkerhet), all miljöpåverkande prestanda (buller etc.) och strukturell
prestanda (ex.vis bärighet, som enbart modelleras via ett åldersmått). För ökad
tilltro till PMS i allmänhet och optimering i synnerhet, bör avvikelserna
analyseras ytterligare och leda till förbättringar vad gäller tillståndsmätning,
tillståndseffekt- & kostnadsmodellering samt matematisk modellering &
implementering.
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Contents
Contents
1 Introduction
1.1 Maintenance problem 1
1.2 Optimisation 3
1.2.1 Simplified optimisation model 4
1.2.2 Lagrangean relaxation 4
1.2.3 Dual optimisation by subgradient methods 6
1.2.4 Dual optimisation by the Dantzig-Wolfe method 7
1.2.5 Subproblem solving by dynamic programming 9
1.3 Program system 11
1.4 Earlier studies 15
1.4.1 Systems based on Markov decision processes (MDPs) 15
1.4.2 The Arizona model 16
1.4.3 The NOS-based PM-systems in Kansas, Alaska and Finland 17
1.4.4 Swedish PMS-based optimisation 18
1.4.5 PMS-systems in Denmark and Norway 18
1.4.6 Other optimisation techniques 18
1.4.7 Decision support systems and integration 19
1.4.8 Survey articles and implementation experience 20
1.5 Study aim and outline 21
1.5.1 Aim of study 21
1.5.2 Outline of the thesis 21
2 Initial study
2.1 Introduction 23
2.2 Model 24
2.2.1 Problem description 24
2.2.2 Mathematical formulation 25
2.3 Method 26
2.3.1 Main procedure 26
2.3.2 Subproblem characteristics 26
2.3.3 Start routine 28
2.3.4 Primal Feasibility Heuristics 29
2.4 Application: PMS- and HIPS-based data 29
2.4.1 Data conversion to OPM 30
2.4.2 Discretization and interpolation experiments 31
2.4.3 Some results 33
2.5 Outlook 36
3 Main study: Problem description
3.1 Road data 37
3.1.1 Road segments 37
3.1.2 Road classes 38
3.1.3 Road constants and pavement data 38
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Contents
3.2 Pavement state 39
3.2.1 State variables 39
3.2.2 State limits 40
3.2.3 IRI-value limits 40
3.2.4 Rut depth limits 40
3.2.5 Age limits 40
3.3 Maintenance 41
3.3.1 Works types 41
3.3.2 Works extent: Layer thickness 41
3.3.3 Restrictions on layer thickness 42
3.4 Traffic effects and costs 42
3.4.1 Travel time cost 42
3.4.2 Vehicle operating cost 42
3.5 Maintenance effects (state transitions) 43
3.5.1 IRI-value immediately after a major maintenance operation, IRI 43
after
3.5.2 IRI-deterioration rate after a major maintenance operation, ΔIRI 43
after
3.5.3 Rut depth immediately after a major maintenance operation, RD 44
after
3.5.4 Deterioration rate of rut depth after a major maint. operation, ΔRD 44
after
3.5.5 Age immediately after a major maintenance operation, Age 44
after
3.5.6 Updated IRI-value, rut depth and age after one year of routine maint. 45
3.6 Maintenance costs 45
3.6.1 Fixed major maintenance costs 46
3.6.2 Variable major maintenance costs 46
3.6.3 Cost of routine maintenance 47
3.7 General data 47
3.7.1 Interest rate 47
3.7.2 Capital scarcity factor 47
3.8 Stochastic submodels 48
3.8.1 Interpolation of state limits 48
3.8.2 Stochastic state transitions 50
3.9 Discussion 52
3.9.1 Limitation 52
3.9.2 Homogeneous segments 53
3.9.3 Robustness 53
3.9.4 Data corrections 54
4 Main study: Basic model and subnet results
4.1 Model 56
4.1.1 Return rate restrictions 56
4.1.2 Segment oriented problem 57
4.1.3 Road class oriented problem 58
4.2 Method 60
4.2.1 Dual optimisation 60
4.2.2 Dual subproblem solving 62
4.2.3 Primal heuristics 64
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