An executive model for network-level pavement maintenance and rehabilitation planning based on linear integer programming

Mahmoud Ameri; Armin Jarrahi

doi:10.3934/jimo.2018179

Article Contents

2020, Volume 16, Issue 2: 795-811. Doi: 10.3934/jimo.2018179

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An executive model for network-level pavement maintenance and rehabilitation planning based on linear integer programming

Mahmoud Ameri^, and
Armin Jarrahi

School of Civil Engineering, Iran University of Science and Technology, Narmak, 16846, Tehran, Iran

Received: October 31, 2017

Revised: June 30, 2018

Early access: December 2018

Published: February 2020

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Abstract

Although having too many details can complicate the planning process, this study involves the formulating of an executive model having a broad range of parameters aimed at network-level pavement maintenance and rehabilitation planning. Four decomposed indicators are used to evaluate the pavement conditions and eight maintenance and rehabilitation categories are defined using these pavement quality indicators. As such, some restrictions called ''technical constraints" are defined to reduce complexity of solving procedure. Using the condition indicators in the form of normalized values and developing technical constraints in a linear integer programming model has improved network level pavement M&R planning. The effectiveness of the developed model was compared by testing it under with-and-without technical constraints conditions over a 3-year planning period in a 10-section road network. It was found that using technical constraints reduced the runtime in resolving the problem by 91%, changed the work plan by 13%, and resulted in a cost increase of 1.2%. Solving runtime reduction issues can be worthwhile in huge networks or long-term planning durations.

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Mathematics Subject Classification: Primary: 90C05, 90C90; Secondary: 90B50.

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Figure 1. Cost comparisons from different objective functions

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Figure 2. Overall cost comparison for different objective functions

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Figure 3. Overall cost comparison in $ ob3 $ with and without technical constraints

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Figure 4. Results of objective functions with and without technical constraints

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Table 1. Comparison of applied methodologies in investigated studies

Study	Optimization Method			Level of Study		Formulation	Model Type	Condition Indicator
Study	MuOb	SiOb	Others	Net	Pro	Formulation	Model Type	Condition Indicator
(TF Fwa, et al., 1996)	√			√		GA	Det	PSI
(T Fwa, et al., 1998)	√			√		GA	Robust	PCI
(Smilowitz & Madanat, 2000)		√		√		LMDP	Condition State	Condition State
(TF Fwa, et al., 2000)	√			√		GA	Robust	PCI
(Ferreira, et al., 2002)		√		√		GA	Det	PSI, IRI, SN
(Chen & Flintsch, 2007)			√		√	LCPA	Fuzzy	PSI, PCI
(Wu, et al., 2008)	√			√		GP & AHP	Det	Condition State
(Abaza & Ashur, 2009)		√		√		CLIP	Det	PCR
(Wu & Flintsch, 2009)	√			√		MDP	Prob	Condition State
(Meneses & Ferreira, 2010)	√			√		GA	Det	PSI
(Moazami, et al., 2011)			√	√		AHP	Fuzzy	PCI
(Irfan, et al., 2012)		√			√	MINLP	Prob	IRI
(Gao & Zhang, 2013)		√		√		Knapsack	Det	PCI
(Medury & Madanat, 2013)		√		√		LP	Prob	Condition State
(Mathew & Isaac, 2014)	√			√		GA	Det	PCI
(Meneses & Ferreira, 2015)	√			√		GA	Det	PSI
(Saha & Ksaibati, 2016)		√		√		LCCA	Det	PSI
(Yepes, et al., 2016)		√		√		GRASP	Det	PCI
(Swei, et al., 2016)		√		√		MINLP	Det & Prob	PCR
Current study	√			√		MILP	Det	$ q \dot{x} (qf, qt, qs, qr), qo $
MuOb - Multi Objective; SiOb - Single Objective; Net - Network; Pro - Project; Det - Deterministic; Prob - Probabilistic

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Table 2. List of indices

Variable Index	Description
$ t, (t \in \{1, 2, \dots, T\}) $	Period (year)
$ n, (n \in \{1, 2, \dots, N\}) $	No. of Section
$ m, (m \in \{1, 2, \dots, M\}) $	M&R actions category
$ b, (b \in \{1, 2, \dots, B\}) $	Auxiliary index corresponding to condition

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Table 3. List of M&R action categories

ID No.	Action Category	Policy
1	Localized safety maintenance	Temporary
2	Localized preventive maintenance	Preventive
3	Level 1: Global preventive maintenance with the objective of improving thermal distresses
4	Level 2: Global preventive maintenance with the objective of improving skid resistance in addition to the level 1 objective
5	Level 3: Global preventive maintenance with the objective of surface irregularity correction in addition to improving the level 2 objective
6	Surface rehabilitation	Corective
7	Deep rehabilitation	(rehabilitation and
8	Reconstruction	reconstruction

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Table 4. Model parameters

Variable	Description	Domain
$ \mu $	A large value (close to infinity)	$ (\mu \to \infty ) $
$\epsilon$	A small value (close to zero)	$ (\epsilon \to 0) $
$va_{n, t}$	Pavement financial value of section $ n $ at the year $ t $ while is in the best condition	$va_{n, t} \in [0, \infty) $
$ drop\dot{x}_{n, t} $	Condition drops from $ t=1 $ to the target year of $ t $ if no M&R action is performed	$drop\dot{x}_{n, t} \in [0, 1] $
$ drop v\dot{x}_{n, m, t, b} $	Condition drops from the year of performing M&R action $ (m) $on a pavement section$ (n) $ with condition index $ b $ to the target year of $ t $	$ drop v\dot{x}_{n, m, t, b} \in [0, 1]$
$ inq\dot{x}_n $	Initial condition	$ inq \dot{x}_n \in [0, 1] $
$ im\dot{x}_m $	Condition improvement	$ im \dot{x}_m \in [0, 1] $
$ crl\dot{x}_n $	Lower threshold of condition	$ crl\dot{x}_n \in[0, 1] $
$ crlo_{n, s} $	Lower threshold of overall condition	$ crlo_{n, s} \in [0, 1] $
$ crh\dot{x}_n $	Upper threshold of condition	$crh\dot{x}_n \in[0, 1] $
$ crho_n $	Upper threshold of overall condition	$ crho_n \in [0, 1] $
$ bu_t $	Allocated budget	$ bu_t \in [0, \infty) $
$ co_{n, m, t} $	M&R action cost (operating cost)	$co_{, m, t} \in [0, \infty) $
$ crq_n $	Critical condition in the vehicle operation cost (VOC) vs overall condition curve	$ crq_n \in [0, 1] $
$ cuc_{n, t} $	VOC at the critical condition	$ cuc_{n, t} \in [0, \infty) $
$cub_{n, t}$	VOC at the worst condition (0)	$cub_{n, t}\in [0, \infty) $
$ cug_{n, t} $	VOC at the best condition (1)	$cug_{n, t}\in [0, \infty) $
$ ce_{n, m, t} $	Delay cost due to performing M&R actions	$ ce_{n, m, t}\in [0, \infty) $
$ k\dot{x} $	Coefficient of $ \dot{x} $ condition in overall condition equation	$ k\dot{x}\in[0, 1] $
c	Deterioration constant in overall condition equation	$ c\in[0, 1] $
$ \dot{x} $ can be replaced by $ f, t, s $ or $ r $ which respectively related to Fatigue, Thermal distress, Skid or Roughness.

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Table 5. Variables

Variable	Description	Domain
$ x_{n, m, t} $	Binary variable for M&R action $ m $, section $ n $ and year $ t $ in the M&R work planning	$ x_{n, m, t} \in \{0, 1\} $
$ q \dot{x}_{n, t} $	Condition indicator	$ q \dot{x}_{n, t} \in [0, 1] $
$ w \dot{x}_{n, t, b} $	Auxiliary variable of condition	$ w \dot{x}_{n, t, b} \in [0.5 -10 \epsilon , 1]$
$ kw \dot{x}_{n, t, b} $	Binary variable determining whether (1) or not (0) auxiliary variable of condition is in index $ b $	$ kw \dot{x}_{n, t, b} \in \{0, 1\} $
$ cubq_{n, t} $	Binary variable determining whether (1) or not (0) the condition is in the poor zone in the VOC vs overall condition curve	$ cubq_{n, t} \in \{0, 1\} $
$ cuqg_{n, t} $	Binary variable determining whether (1) or not (0) the condition is in the good zone in the VOC vs overall condition curve	$ cuqg_{n, t} \in \{0, 1\} $
$ cb_{n, t} $	Auxiliary variable for the effect of poor condition on the VOC	$ cb_{n, t} \in [0, 1] $
$ cg_{n, t} $	Auxiliary variable for the effect of good condition on the VOC	$cg_{n, t} \in [0, 1] $
$ qo_{n, t} $	Overall condition indicator	$ qo_{n, t} \in [0, 1] $
$ k_{n, m, t} $	Intensity of distress variable	$ k_{n, m, t} \in [0, 1] $
$ z_{n, m, t} $	Auxiliary variable for intensity of distress	$ z_{n, m, t} \in [0, 1] $
$ da \dot{x}_{n, t} $	Condition drop	$ da \dot{x}_{n, t} \in [0, 1] $
$ dd \dot{x}_{n, t} $	Auxiliary variable for condition drop	$ dd \dot{x}_{n, t} \in [0, 1] $
$ d \dot{x}_{n, m, t, t^{'}} $	Condition drop at $ t^{'} $ once M&R action was performed at $ t $	$ {d \dot{x}_{n, m, t, t^{'}} } \in [0, 1] $
$ \dot{x} $ can be replaced by $ f, t, s $ or $ r $ which respectively related to Fatigue, Thermal distress, Skid or Roughness.

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Table 6. Types of influential costs used in this study

ID	Cost
$ ce $	Delay cost due to performing M&R actions
$ co $	Operating cost
$ cu $	Vehicle Operation cost
$ va $	Consumed pavement financial value compared to the highest pavement value

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Table 7. Limits of condition indicators for ignoring M&R action

m	1	2	3	4	5	6	7	8
$ qo $	$ crl< $	$ <crl $ ^*	$ <crl $	$ <crl $	$ <crl $	-	-	$ **crh< $
$ qf $	-	$ <crl$	$ <crl$	$<crl$	$<crl$	-	-	-
$ qt $	-	-	$ crh< $	-	-	-	-	-
$ qs $	-	-	-	$ crh< $	-	-	-	-
$ qr $	-	-	-	-	$ crh< $	-	-	-
^* The $ <crl $ indicates the lower threshold value for a condition indicator in which performing M&R action over sections with values of less than that is not justified. ^** The $crh < $ indicates the upper threshold value for a condition indicator in which performing M&R action over sections with values greater than that is not justified. - No limitations on performing M&R action.

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Table 8. M&R actions assignment results

$ n $ (section)	$ ob1 $			$ ob2 $			$ ob3 $
$ n $ (section)	$ t=1 $	$ t=2 $	$ t=3$	$ t=1 $	$ t=2 $	$ t=3 $	$ t=1 $	$ t=2$	$ t=3 $
1	Noting	6	2	Noting	Noting	Noting	Noting	6	2
2	5	5	6	Noting	Noting	Noting	Noting	6	2
3	6	6	2	6	Noting	Noting	6	4	Noting
4	6	2	2	6	2	Noting	6	2	Noting
5	Noting	6	6	6	Noting	Noting	6	Noting	Noting
6	5	2	2, 5	5	2	Noting	5	2	2
7	6	5	6	6	Noting	Noting	6	2	Noting
8	Noting	6	2, 5	6	2	Noting	6	2	Noting
9	4	2, 4	6	2, 4	2	Noting	2, 4	2	Noting
10	2	5	2	Noting	Noting	Noting	Noting	Noting	2

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Table 9. Percentage of allocation to the available budget in each year

$ t $ (year)	Budget Utilization (%)			Budget (＄1000)
$ t $ (year)	$ob1 $	$ob2$	$ob3 $	Budget (＄1000)
1	99.94	99.97	99.97	2518
2	99.74	6.85	80.68	2946
3	99.99	0	3.37	3450
Total	99.89	30.50	56.21	8914

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Table 10. With and without technical constraints comparison of M&R action assignments

$ n $	$ ob3 $			$ ob3 $ Without
$ n $	$ t=1 $	$ t=2 $	$ t=3 $	$ t=1 $	$ t=2 $	$ t=3 $
1	Noting	6	2	Noting	6	2
2	Noting	6	2	Noting	6	2
3	6	4	Noting	6	4	Noting
4	6	2	Noting	6	2	Noting
5	6	Noting	Noting	6	Noting	Noting
6	5	2	2	2, 5	2	2
7	6	2	Noting	4	6	Noting
8	6	2	Noting	6	2	Noting
9	2, 4	2	Noting	2, 4	2	Noting
10	Noting	Noting	2	Noting	5	2

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Table 11. With and without technical constraints runtime comparisons

Objective Function	$ ob1 $	$ ob2 $	$ ob3 $	$ ob3 $ Without
Solution Time (min)	46	1	9	108

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