American Institute of Mathematical Sciences

Advanced
doi: 10.3934/jimo.2018179

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  November 2017 Revised  July 2018 Published  December 2018

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.

Keywords: Optimization, M&R, network level planning, pavement management, linear integer programming.
Mathematics Subject Classification: Primary: 90C05, 90C90; Secondary: 90B50.
Citation: Mahmoud Ameri, Armin Jarrahi. An executive model for network-level pavement maintenance and rehabilitation planning based on linear integer programming. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018179
References:
[1]

K. A. Abaza and S. A. Ashur, Optimum microscopic pavement management model using constrained integer linear programming, International Journal of Pavement Engineering, 10 (2009), 149-160.   Google Scholar

[2]

C. Chen and G. Flintsch, Fuzzy logic pavement maintenance and rehabilitation triggering approach for probabilistic life-cycle cost analysis, Transportation Research Record: Journal of the Transportation Research Board, 1990 (2007), 80-91.   Google Scholar

[3]

A. Ferreira, L. Picado-Santos and A. Antunes, A segment-linked optimization model for deterministic pavement management systems, International Journal of Pavement Engineering, 3 (2002), 95-105.   Google Scholar

[4]

T. F. Fwa, W. T. Chan and C. Y. Tan, Genetic-algorithm programming of road maintenance and rehabilitation, Journal of Transportation Engineering, 122 (1996), 246-253.   Google Scholar

[5]

T. Fwa, W. Chan and K. Hoque, Analysis of pavement management activities programming by genetic algorithms, Transportation Research Record: Journal of the Transportation Research Board, 1643 (1998), 1-6.   Google Scholar

[6]

T. F. Fwa, W. T. Chan and K. Z. Hoque, Multiobjective optimization for pavement maintenance programming, Journal of Transportation Engineering, 126 (2000), 367-374.   Google Scholar

[7]

L. Gao and Z. Zhang, Management of pavement maintenance, rehabilitation, and reconstruction through network partition, Transportation Research Record: Journal of the Transportation Research Board, 2366 (2013), 59-63.   Google Scholar

[8]

G. Chiandussi, M. Codegone, S. Ferrero and F. E. Varesio, Comparison of multi-objective optimization methodologies for engineering applications, Computers & Mathematics with Applications, 63 (2012), 912-942.  doi: 10.1016/j.camwa.2011.11.057.  Google Scholar

[9]

M. Irfan, M. B. Khurshid, Q. Bai, S. Labi and T. L. Morin, Establishing optimal project-level strategies for pavement maintenance and rehabilitation-a framework and case study, Engineering Optimization, 44 (2012), 565-589.   Google Scholar

[10]

J. Mallela and S. Sadavisam, Work Zone Road User Costs: Concepts and Applications, US Department of Transportation, Federal Highway Administration, 2011. Google Scholar

[11]

A. Manik, K. Gopalakrishnan, A. Singh and S. Yan, Neural networks surrogate models for simulating payment risk in pavement construction, Journal of Civil Engineering and Management, 14 (2008), 235-240.   Google Scholar

[12]

B. S. Mathew and K. P. Isaac, Optimisation of maintenance strategy for rural road network using genetic algorithm, International Journal of Pavement Engineering, 15 (2014), 352-360.   Google Scholar

[13]

A. Medury and S. Madanat, Simultaneous network optimization approach for pavement management systems, Journal of Infrastructure Systems, 20 (2013), 04014010. Google Scholar

[14]

S. Meneses and A. Ferreira, Multi-objective decision-aid tool for pavement management systems, Proceedings of the 12th World Conference on Transport Research, (2010), 1-11.   Google Scholar

[15]

S. Meneses and A. Ferreira, Flexible pavement maintenance programming considering the minimisation of maintenance and rehabilitation costs and the maximisation of the residual value of pavements, International Journal of Pavement Engineering, 16 (2015), 571-586.   Google Scholar

[16]

D. Moazami, H. Behbahani and R. Muniandy, Pavement rehabilitation and maintenance prioritization of urban roads using fuzzy logic, Expert Systems with Applications, 38 (2011), 12869-12879.   Google Scholar

[17]

D. Moazami and R. Muniandy, Fuzzy inference and multi-criteria decision making applications in pavement rehabilitation prioritization, Australian Journal of Basic and Applied Sciences, 4 (2010), 4740-4748.   Google Scholar

[18]

M. S. Pishvaee and J. Razmi, Environmental supply chain network design using multi-objective fuzzy mathematical programming, Applied Mathematical Modelling, 36 (2012), 3433-3446.  doi: 10.1016/j.apm.2011.10.007.  Google Scholar

[19]

M. Ramezani, M. Bashiri and R. Tavakkoli-Moghaddam, A robust design for a closed-loop supply chain network under an uncertain environment, The International Journal of Advanced Manufacturing Technology, 66 (2013), 825-843.   Google Scholar

[20]

P. Saha and K. Ksaibati, A risk-based optimisation methodology for pavement management system of county roads, International Journal of Pavement Engineering, 17 (2016), 913-923.   Google Scholar

[21]

T. Scheinberg and P. Ch Anastasopoulos, Pavement preservation programming: A multi-year multi-constraint optimization methodology, In presentation at the 89th Annual Meeting of the Transportation Research Board and publication in the Transportation Research Record, 2010. Google Scholar

[22]

M. Y. Shahin, Pavement management for airports, Roads and Parking lots, 2005. Google Scholar

[23]

K. Smilowitz and S. Madanat, Optimal inspection and maintenance policies for infrastructure networks, Computer-Aided Civil and Infrastructure Engineering, 15 (2000), 5-13.   Google Scholar

[24]

O. Swei, J. Gregory and R. Kirchain, Pavement management systems: Opportunities to improve the current frameworks, In TRB 2016 Annual Meeting, volume 16, 2015. Google Scholar

[25]

F. Wang, Z. Zhang and R. Machemehl, Decision-making problem for managing pavement maintenance and rehabilitation projects, Transportation Research Record: Journal of the Transportation Research Board, 1853 (2003), 21-28.   Google Scholar

[26]

Z. Wu, G. W. Flintsch and T. Chowdhury, Hybrid multiobjective optimization model for regional pavement-preservation resource allocation, Transportation Research Record, 2084 (2008), 28-37.   Google Scholar

[27]

Z. Wu and G. W. Flintsch, Pavement preservation optimization considering multiple objectives and budget variability, Journal of Transportation Engineering, 135 (2009), 305-315.   Google Scholar

[28]

V. Yepes, C. Torres-Machi, A. Chamorro and E. Pellicer, Optimal pavement maintenance programs based on a hybrid greedy randomized adaptive search procedure algorithm, Journal of Civil Engineering and Management, 22 (2016), 540-550.   Google Scholar

show all references

References:
[1]

K. A. Abaza and S. A. Ashur, Optimum microscopic pavement management model using constrained integer linear programming, International Journal of Pavement Engineering, 10 (2009), 149-160.   Google Scholar

[2]

C. Chen and G. Flintsch, Fuzzy logic pavement maintenance and rehabilitation triggering approach for probabilistic life-cycle cost analysis, Transportation Research Record: Journal of the Transportation Research Board, 1990 (2007), 80-91.   Google Scholar

[3]

A. Ferreira, L. Picado-Santos and A. Antunes, A segment-linked optimization model for deterministic pavement management systems, International Journal of Pavement Engineering, 3 (2002), 95-105.   Google Scholar

[4]

T. F. Fwa, W. T. Chan and C. Y. Tan, Genetic-algorithm programming of road maintenance and rehabilitation, Journal of Transportation Engineering, 122 (1996), 246-253.   Google Scholar

[5]

T. Fwa, W. Chan and K. Hoque, Analysis of pavement management activities programming by genetic algorithms, Transportation Research Record: Journal of the Transportation Research Board, 1643 (1998), 1-6.   Google Scholar

[6]

T. F. Fwa, W. T. Chan and K. Z. Hoque, Multiobjective optimization for pavement maintenance programming, Journal of Transportation Engineering, 126 (2000), 367-374.   Google Scholar

[7]

L. Gao and Z. Zhang, Management of pavement maintenance, rehabilitation, and reconstruction through network partition, Transportation Research Record: Journal of the Transportation Research Board, 2366 (2013), 59-63.   Google Scholar

[8]

G. Chiandussi, M. Codegone, S. Ferrero and F. E. Varesio, Comparison of multi-objective optimization methodologies for engineering applications, Computers & Mathematics with Applications, 63 (2012), 912-942.  doi: 10.1016/j.camwa.2011.11.057.  Google Scholar

[9]

M. Irfan, M. B. Khurshid, Q. Bai, S. Labi and T. L. Morin, Establishing optimal project-level strategies for pavement maintenance and rehabilitation-a framework and case study, Engineering Optimization, 44 (2012), 565-589.   Google Scholar

[10]

J. Mallela and S. Sadavisam, Work Zone Road User Costs: Concepts and Applications, US Department of Transportation, Federal Highway Administration, 2011. Google Scholar

[11]

A. Manik, K. Gopalakrishnan, A. Singh and S. Yan, Neural networks surrogate models for simulating payment risk in pavement construction, Journal of Civil Engineering and Management, 14 (2008), 235-240.   Google Scholar

[12]

B. S. Mathew and K. P. Isaac, Optimisation of maintenance strategy for rural road network using genetic algorithm, International Journal of Pavement Engineering, 15 (2014), 352-360.   Google Scholar

[13]

A. Medury and S. Madanat, Simultaneous network optimization approach for pavement management systems, Journal of Infrastructure Systems, 20 (2013), 04014010. Google Scholar

[14]

S. Meneses and A. Ferreira, Multi-objective decision-aid tool for pavement management systems, Proceedings of the 12th World Conference on Transport Research, (2010), 1-11.   Google Scholar

[15]

S. Meneses and A. Ferreira, Flexible pavement maintenance programming considering the minimisation of maintenance and rehabilitation costs and the maximisation of the residual value of pavements, International Journal of Pavement Engineering, 16 (2015), 571-586.   Google Scholar

[16]

D. Moazami, H. Behbahani and R. Muniandy, Pavement rehabilitation and maintenance prioritization of urban roads using fuzzy logic, Expert Systems with Applications, 38 (2011), 12869-12879.   Google Scholar

[17]

D. Moazami and R. Muniandy, Fuzzy inference and multi-criteria decision making applications in pavement rehabilitation prioritization, Australian Journal of Basic and Applied Sciences, 4 (2010), 4740-4748.   Google Scholar

[18]

M. S. Pishvaee and J. Razmi, Environmental supply chain network design using multi-objective fuzzy mathematical programming, Applied Mathematical Modelling, 36 (2012), 3433-3446.  doi: 10.1016/j.apm.2011.10.007.  Google Scholar

[19]

M. Ramezani, M. Bashiri and R. Tavakkoli-Moghaddam, A robust design for a closed-loop supply chain network under an uncertain environment, The International Journal of Advanced Manufacturing Technology, 66 (2013), 825-843.   Google Scholar

[20]

P. Saha and K. Ksaibati, A risk-based optimisation methodology for pavement management system of county roads, International Journal of Pavement Engineering, 17 (2016), 913-923.   Google Scholar

[21]

T. Scheinberg and P. Ch Anastasopoulos, Pavement preservation programming: A multi-year multi-constraint optimization methodology, In presentation at the 89th Annual Meeting of the Transportation Research Board and publication in the Transportation Research Record, 2010. Google Scholar

[22]

M. Y. Shahin, Pavement management for airports, Roads and Parking lots, 2005. Google Scholar

[23]

K. Smilowitz and S. Madanat, Optimal inspection and maintenance policies for infrastructure networks, Computer-Aided Civil and Infrastructure Engineering, 15 (2000), 5-13.   Google Scholar

[24]

O. Swei, J. Gregory and R. Kirchain, Pavement management systems: Opportunities to improve the current frameworks, In TRB 2016 Annual Meeting, volume 16, 2015. Google Scholar

[25]

F. Wang, Z. Zhang and R. Machemehl, Decision-making problem for managing pavement maintenance and rehabilitation projects, Transportation Research Record: Journal of the Transportation Research Board, 1853 (2003), 21-28.   Google Scholar

[26]

Z. Wu, G. W. Flintsch and T. Chowdhury, Hybrid multiobjective optimization model for regional pavement-preservation resource allocation, Transportation Research Record, 2084 (2008), 28-37.   Google Scholar

[27]

Z. Wu and G. W. Flintsch, Pavement preservation optimization considering multiple objectives and budget variability, Journal of Transportation Engineering, 135 (2009), 305-315.   Google Scholar

[28]

V. Yepes, C. Torres-Machi, A. Chamorro and E. Pellicer, Optimal pavement maintenance programs based on a hybrid greedy randomized adaptive search procedure algorithm, Journal of Civil Engineering and Management, 22 (2016), 540-550.   Google Scholar

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
StudyOptimization MethodLevel of StudyFormulationModel TypeCondition Indicator
MuObSiObOthersNetPro
(TF Fwa, et al., 1996)√√GADetPSI
(T Fwa, et al., 1998)√√GARobustPCI
(Smilowitz & Madanat, 2000)√√LMDPCondition StateCondition State
(TF Fwa, et al., 2000)√√GARobustPCI
(Ferreira, et al., 2002)√√GADetPSI, IRI, SN
(Chen & Flintsch, 2007)√√LCPAFuzzyPSI, PCI
(Wu, et al., 2008)√√GP & AHPDetCondition State
(Abaza & Ashur, 2009)√√CLIPDetPCR
(Wu & Flintsch, 2009)√√MDPProbCondition State
(Meneses & Ferreira, 2010)√√GADetPSI
(Moazami, et al., 2011)√√AHPFuzzyPCI
(Irfan, et al., 2012)√√MINLPProbIRI
(Gao & Zhang, 2013)√√KnapsackDetPCI
(Medury & Madanat, 2013)√√LPProbCondition State
(Mathew & Isaac, 2014)√√GADetPCI
(Meneses & Ferreira, 2015)√√GADetPSI
(Saha & Ksaibati, 2016)√√LCCADetPSI
(Yepes, et al., 2016)√√GRASPDetPCI
(Swei, et al., 2016)√√MINLPDet & ProbPCR
Current study√√MILPDet$ 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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StudyOptimization MethodLevel of StudyFormulationModel TypeCondition Indicator
MuObSiObOthersNetPro
(TF Fwa, et al., 1996)√√GADetPSI
(T Fwa, et al., 1998)√√GARobustPCI
(Smilowitz & Madanat, 2000)√√LMDPCondition StateCondition State
(TF Fwa, et al., 2000)√√GARobustPCI
(Ferreira, et al., 2002)√√GADetPSI, IRI, SN
(Chen & Flintsch, 2007)√√LCPAFuzzyPSI, PCI
(Wu, et al., 2008)√√GP & AHPDetCondition State
(Abaza & Ashur, 2009)√√CLIPDetPCR
(Wu & Flintsch, 2009)√√MDPProbCondition State
(Meneses & Ferreira, 2010)√√GADetPSI
(Moazami, et al., 2011)√√AHPFuzzyPCI
(Irfan, et al., 2012)√√MINLPProbIRI
(Gao & Zhang, 2013)√√KnapsackDetPCI
(Medury & Madanat, 2013)√√LPProbCondition State
(Mathew & Isaac, 2014)√√GADetPCI
(Meneses & Ferreira, 2015)√√GADetPSI
(Saha & Ksaibati, 2016)√√LCCADetPSI
(Yepes, et al., 2016)√√GRASPDetPCI
(Swei, et al., 2016)√√MINLPDet & ProbPCR
Current study√√MILPDet$ q \dot{x} (qf, qt, qs, qr), qo $
MuOb - Multi Objective; SiOb - Single Objective; Net - Network; Pro - Project; Det - Deterministic; Prob - Probabilistic
Table 2.  List of indices
Variable IndexDescription
$ 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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Variable IndexDescription
$ 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
Table 3.  List of M&R action categories
ID No.Action CategoryPolicy
1Localized safety maintenanceTemporary
2Localized preventive maintenancePreventive
3Level 1: Global preventive maintenance with the objective of improving thermal distresses
4Level 2: Global preventive maintenance with the objective of improving skid resistance in addition to the level 1 objective
5Level 3: Global preventive maintenance with the objective of surface irregularity correction in addition to improving the level 2 objective
6Surface rehabilitationCorective
7Deep rehabilitation(rehabilitation and
8Reconstructionreconstruction
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ID No.Action CategoryPolicy
1Localized safety maintenanceTemporary
2Localized preventive maintenancePreventive
3Level 1: Global preventive maintenance with the objective of improving thermal distresses
4Level 2: Global preventive maintenance with the objective of improving skid resistance in addition to the level 1 objective
5Level 3: Global preventive maintenance with the objective of surface irregularity correction in addition to improving the level 2 objective
6Surface rehabilitationCorective
7Deep rehabilitation(rehabilitation and
8Reconstructionreconstruction
Table 4.  Model parameters
VariableDescriptionDomain
$ \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 $$ drop v\dot{x}_{n, m, t, b} \in [0, 1]$
$ inq\dot{x}_n $Initial condition$ inq \dot{x}_n \in [0, 1] $
$ im\dot{x}_m $Condition improvement$ im \dot{x}_m \in [0, 1] $
$ crl\dot{x}_n $Lower threshold of condition$ crl\dot{x}_n \in[0, 1] $
$ crlo_{n, s} $Lower threshold of overall condition$ crlo_{n, s} \in [0, 1] $
$ crh\dot{x}_n $Upper threshold of condition$crh\dot{x}_n \in[0, 1] $
$ crho_n $Upper threshold of overall condition$ crho_n \in [0, 1] $
$ bu_t $Allocated budget$ bu_t \in [0, \infty) $
$ co_{n, m, t} $M&R action cost (operating cost) $co_{, m, t} \in [0, \infty) $
$ crq_n $Critical condition in the vehicle operation cost (VOC) vs overall condition curve$ crq_n \in [0, 1] $
$ cuc_{n, t} $VOC at the critical condition$ cuc_{n, t} \in [0, \infty) $
$cub_{n, t}$VOC at the worst condition (0)$cub_{n, t}\in [0, \infty) $
$ cug_{n, t} $VOC at the best condition (1)$cug_{n, t}\in [0, \infty) $
$ ce_{n, m, t} $Delay cost due to performing M&R actions$ ce_{n, m, t}\in [0, \infty) $
$ k\dot{x} $Coefficient of $ \dot{x} $ condition in overall condition equation$ k\dot{x}\in[0, 1] $
cDeterioration constant in overall condition equation$ c\in[0, 1] $
$ \dot{x} $ can be replaced by $ f, t, s $ or $ r $ which respectively related to Fatigue, Thermal distress, Skid or Roughness.
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VariableDescriptionDomain
$ \mu $A large value (close to infinity)$ (\mu \to \infty ) $
$\epsilon$A small value (close to zero)$ (\epsilon \to 0) $
$va_{n, t}$Pavement financial value of section $ n $ at the year $ t $ while is in the best condition$va_{n, t} \in [0, \infty) $
$ drop\dot{x}_{n, t} $Condition drops from $ t=1 $ to the target year of $ t $ if no M&R action is performed$drop\dot{x}_{n, t} \in [0, 1] $
$ drop v\dot{x}_{n, m, t, b} $Condition drops from the year of performing M&R action $ (m) $on a pavement section$ (n) $ with condition index $ b $ to the target year of $ t $$ 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] $
cDeterioration 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.
Table 5.  Variables
VariableDescriptionDomain
$ 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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VariableDescriptionDomain
$ 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.
Table 6.  Types of influential costs used in this study
IDCost
$ 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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IDCost
$ 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
Table 7.  Limits of condition indicators for ignoring M&R action
m12345678
$ qo $ $ crl< $ $ <crl $ * $ <crl $ $ <crl $ $ <crl $--$ **crh< $
$ qf $-$ <crl$$ <crl$$<crl$$<crl$---
$ qt $--$ crh< $-----
$ qs $---$ crh< $----
$ qr $----$ crh< $---
* The $ <crl $ indicates the lower threshold value for a condition indicator in which performing M&R action over sections with values of less than that is not justified.
** The $crh < $ indicates the upper threshold value for a condition indicator in which performing M&R action over sections with values greater than that is not justified.
- No limitations on performing M&R action.
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m12345678
$ qo $ $ crl< $ $ <crl $ * $ <crl $ $ <crl $ $ <crl $--$ **crh< $
$ qf $-$ <crl$$ <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.
Table 8.  M&R actions assignment results
$ n $
(section)		$ ob1 $$ ob2 $$ ob3 $
$ t=1 $$ t=2 $$ t=3$$ t=1 $$ t=2 $$ t=3 $$ t=1 $$ t=2$$ t=3 $
1Noting62NotingNotingNotingNoting62
2556NotingNotingNotingNoting62
36626NotingNoting64Noting
462262Noting62Noting
5Noting666NotingNoting6NotingNoting
6522, 552Noting522
76566NotingNoting62Noting
8Noting62, 562Noting62Noting
942, 462, 42Noting2, 42Noting
10252NotingNotingNotingNotingNoting2
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$ n $
(section)		$ ob1 $$ ob2 $$ ob3 $
$ t=1 $$ t=2 $$ t=3$$ t=1 $$ t=2 $$ t=3 $$ t=1 $$ t=2$$ t=3 $
1Noting62NotingNotingNotingNoting62
2556NotingNotingNotingNoting62
36626NotingNoting64Noting
462262Noting62Noting
5Noting666NotingNoting6NotingNoting
6522, 552Noting522
76566NotingNoting62Noting
8Noting62, 562Noting62Noting
942, 462, 42Noting2, 42Noting
10252NotingNotingNotingNotingNoting2
Table 9.  Percentage of allocation to the available budget in each year
$ t $ (year)Budget Utilization (%)Budget (＄1000)
$ob1 $ $ob2$ $ob3 $
199.9499.9799.972518
299.746.8580.682946
399.9903.373450
Total99.8930.5056.218914
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$ t $ (year)Budget Utilization (%)Budget (＄1000)
$ob1 $ $ob2$ $ob3 $
199.9499.9799.972518
299.746.8580.682946
399.9903.373450
Total99.8930.5056.218914
Table 10.  With and without technical constraints comparison of M&R action assignments
$ n $$ ob3 $$ ob3 $ Without
$ t=1 $$ t=2 $$ t=3 $$ t=1 $$ t=2 $$ t=3 $
1Noting62Noting62
2Noting62Noting62
364Noting64Noting
462Noting62Noting
56NotingNoting6NotingNoting
65222, 522
762Noting46Noting
862Noting62Noting
92, 42Noting2, 42Noting
10NotingNoting2Noting52
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$ n $$ ob3 $$ ob3 $ Without
$ t=1 $$ t=2 $$ t=3 $$ t=1 $$ t=2 $$ t=3 $
1Noting62Noting62
2Noting62Noting62
364Noting64Noting
462Noting62Noting
56NotingNoting6NotingNoting
65222, 522
762Noting46Noting
862Noting62Noting
92, 42Noting2, 42Noting
10NotingNoting2Noting52
Table 11.  With and without technical constraints runtime comparisons
Objective Function$ ob1 $$ ob2 $$ ob3 $$ ob3 $ Without
Solution Time (min)4619108
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Objective Function$ ob1 $$ ob2 $$ ob3 $$ ob3 $ Without
Solution Time (min)4619108
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