# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2018179

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

 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.

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
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Cost comparisons from different objective functions
Overall cost comparison for different objective functions
Overall cost comparison in $ob3$ with and without technical constraints
Results of objective functions with and without technical constraints
Comparison of applied methodologies in investigated studies
 Study Optimization Method Level of Study Formulation Model Type Condition Indicator MuOb SiOb Others Net Pro (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
 Study Optimization Method Level of Study Formulation Model Type Condition Indicator MuOb SiOb Others Net Pro (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
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
 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
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
 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
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.
 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.
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.
 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.
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
 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
Limits of condition indicators for ignoring M&R action
 m 1 2 3 4 5 6 7 8 $qo$ $crl<$ $ m 1 2 3 4 5 6 7 8$ qo  crl< 
M&R actions assignment results
 $n$(section) $ob1$ $ob2$ $ob3$ $t=1$ $t=2$ $t=3$ $t=1$ $t=2$ $t=3$ $t=1$ $t=2$ $t=3$ 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
 $n$(section) $ob1$ $ob2$ $ob3$ $t=1$ $t=2$ $t=3$ $t=1$ $t=2$ $t=3$ $t=1$ $t=2$ $t=3$ 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
Percentage of allocation to the available budget in each year
 $t$ (year) Budget Utilization (%) Budget (＄1000) $ob1$ $ob2$ $ob3$ 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
 $t$ (year) Budget Utilization (%) Budget (＄1000) $ob1$ $ob2$ $ob3$ 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
With and without technical constraints comparison of M&R action assignments
 $n$ $ob3$ $ob3$ Without $t=1$ $t=2$ $t=3$ $t=1$ $t=2$ $t=3$ 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
 $n$ $ob3$ $ob3$ Without $t=1$ $t=2$ $t=3$ $t=1$ $t=2$ $t=3$ 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
With and without technical constraints runtime comparisons
 Objective Function $ob1$ $ob2$ $ob3$ $ob3$ Without Solution Time (min) 46 1 9 108
 Objective Function $ob1$ $ob2$ $ob3$ $ob3$ Without Solution Time (min) 46 1 9 108
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