3.2. Model calibration
In this section, a model-based calibration method [33] is used
for estimating appropriate values for unknown parameters of the
SD model. The units under analysis in this model are defined as
road sections. By modeling multiple road sections, analysis of
condition of highway network is also possible. Table 1 shows the
road condition data over the years 2002 to 2008 for the fourteen
road sections in 17 miles of the highway network under analysis
after some data cleaning. These data points are used for the
calibration process. The cells that are bold, underlined, and italic
show the years in which preventive, corrective, and restorative
maintenance operations have been performed on the road sections
under analysis, respectively. Preventive maintenance (PM) preserves
the existing pavement integrity and reduces the rate of
deterioration. Corrective maintenance (CM) maintains the characteristics
and structural integrity of an existing pavement for
continued serviceability. Restorative maintenance (RM) refers to
new surface layers that restore the pavement structure to a level
similar to the original condition of the pavement [34,35]. The
maintenance operations in the calibration runs happen exogenously
according to the data provided in Table 1. As the empirical
data shows, after performing maintenance operations on road
sections under analysis, the condition of the road sections have
gone back to perfect condition in the next year. To mimic the real
world scenarios, when a maintenance operation is performed
(according to the schedule observed in Table 1) in the calibration,
the level of “Road Quality” is also restored to its maximum
possible values which, consequently, lead to maximum possible
CCI (i.e., CCI¼100). Note that in real world, we may also observe
scenarios in which the road condition is not restored to perfect
condition after performing a maintenance operation (for example
if we do not perform the right maintenance operation given the
level of deterioration of the road).
The objective of the calibration process is to estimate the values
of the unknown parameters that capture the deterioration rate of
road sections (i.e., the rate of depletion of the Road Quality stock)
in the SD model so that the CCI of the simulated road sections
follow the CCI data provided in Table 1. Thus, an error term is
defined for each road section as the difference between the road
section’s CCI obtained from the model and the corresponding CCI
available from the Table 1. The payoff function of the calibration
process (that needs to be minimized) is then defined as the
weighted sum of fourteen squared error terms corresponding to
the fourteen road sections. Error terms are weighted proportional
to the inverse of the standard deviation of the error terms, which
leads to maximum likelihood estimates for the estimated parameters
assuming normally distributed errors [36]. All simulations
and calibration were conducted in Vensim[37] software using
Euler integration method with time step 0.0625 year (results
showed no sensitivity to smaller time steps). Modified Powell
conjugate search [38] was used for calibration. Calibration was
restarted multiple times from different points of parameter space
to increase confidence in finding the global peak in the numerical
optimization.
As an example, Fig. 5 shows the comparison of CCI data
obtained from the Table 1 and from the SD model for the road
section “1” and “12”.
The unknown parameters of the model have been estimated so
that the model mimics the deterioration process that happened in
the highway network under analysis. Table 2 shows the estimated
values for model parameters obtained from calibration process. All
parameters are found to take realistic values. The allowable load
cycles, approximately at 100 million, is comparable with the
relevant load cycles used in highway design. The estimates show
a modest increase in deterioration with age (Beta¼1.42) and a
faster deterioration rate with reduced quality (Delta¼2), leading
to a notable reinforcing loop, where allowing the road section to
go into a low-quality region leads to more significant costs of
deterioration.
3.2. Model calibration
In this section, a model-based calibration method [33] is used
for estimating appropriate values for unknown parameters of the
SD model. The units under analysis in this model are defined as
road sections. By modeling multiple road sections, analysis of
condition of highway network is also possible. Table 1 shows the
road condition data over the years 2002 to 2008 for the fourteen
road sections in 17 miles of the highway network under analysis
after some data cleaning. These data points are used for the
calibration process. The cells that are bold, underlined, and italic
show the years in which preventive, corrective, and restorative
maintenance operations have been performed on the road sections
under analysis, respectively. Preventive maintenance (PM) preserves
the existing pavement integrity and reduces the rate of
deterioration. Corrective maintenance (CM) maintains the characteristics
and structural integrity of an existing pavement for
continued serviceability. Restorative maintenance (RM) refers to
new surface layers that restore the pavement structure to a level
similar to the original condition of the pavement [34,35]. The
maintenance operations in the calibration runs happen exogenously
according to the data provided in Table 1. As the empirical
data shows, after performing maintenance operations on road
sections under analysis, the condition of the road sections have
gone back to perfect condition in the next year. To mimic the real
world scenarios, when a maintenance operation is performed
(according to the schedule observed in Table 1) in the calibration,
the level of “Road Quality” is also restored to its maximum
possible values which, consequently, lead to maximum possible
CCI (i.e., CCI¼100). Note that in real world, we may also observe
scenarios in which the road condition is not restored to perfect
condition after performing a maintenance operation (for example
if we do not perform the right maintenance operation given the
level of deterioration of the road).
The objective of the calibration process is to estimate the values
of the unknown parameters that capture the deterioration rate of
road sections (i.e., the rate of depletion of the Road Quality stock)
in the SD model so that the CCI of the simulated road sections
follow the CCI data provided in Table 1. Thus, an error term is
defined for each road section as the difference between the road
section’s CCI obtained from the model and the corresponding CCI
available from the Table 1. The payoff function of the calibration
process (that needs to be minimized) is then defined as the
weighted sum of fourteen squared error terms corresponding to
the fourteen road sections. Error terms are weighted proportional
to the inverse of the standard deviation of the error terms, which
leads to maximum likelihood estimates for the estimated parameters
assuming normally distributed errors [36]. All simulations
and calibration were conducted in Vensim[37] software using
Euler integration method with time step 0.0625 year (results
showed no sensitivity to smaller time steps). Modified Powell
conjugate search [38] was used for calibration. Calibration was
restarted multiple times from different points of parameter space
to increase confidence in finding the global peak in the numerical
optimization.
As an example, Fig. 5 shows the comparison of CCI data
obtained from the Table 1 and from the SD model for the road
section “1” and “12”.
The unknown parameters of the model have been estimated so
that the model mimics the deterioration process that happened in
the highway network under analysis. Table 2 shows the estimated
values for model parameters obtained from calibration process. All
parameters are found to take realistic values. The allowable load
cycles, approximately at 100 million, is comparable with the
relevant load cycles used in highway design. The estimates show
a modest increase in deterioration with age (Beta¼1.42) and a
faster deterioration rate with reduced quality (Delta¼2), leading
to a notable reinforcing loop, where allowing the road section to
go into a low-quality region leads to more significant costs of
deterioration.
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