True optimization of pavement maintenance options with what-If models

, −0.57, −0.16, 0.00,
−0.19, 0.00]
0.0 83.96 0.63
3 [4, 8, 13, 15, 19, 21; 6, 3, 3, 3, 3, 3] 83.96 0.00
k = 9 0 [4, 7, 10,12, 14, 16, 18, 20, 22; 6, 3, 3,
3, 3, 3, 3, 3, 3]
[−0.98, 1.85, 0.15, −0.25, 0.10, −0.04,
−0.02, 0.05, 0.00; −0.86, −0.92,
−0.74, 0.00, −0.54, 0.00, −0.41,
0.00, −0.41]
0.4 81.60
1 [4, 8, 10,12, 14, 16, 18, 20, 22; 6, 3, 3,
3, 3, 3, 3, 3, 3]
[−3.16, −0.93, −1.56, 0.07, −1.32,
0.00, −1.18, −0.16, 0.00; −0.71,
−0.79, 0.00, −0.60, −0.02, −0.46,
0.00, −0.41, 0.00]
0.0 83.45 2.26
2 [4, 8, 10,12, 14, 16, 18, 20, 22; 6, 3, 3,
3, 3, 3, 3, 3, 3]
83.45 0.00
k = 13 0 [4, 6, 8, 10, 12, 14, 16, 17, 18, 19, 20,
21, 22; 6, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
3, 3]
[−3.16, −0.71, 0.06, 0.00, 0.03, −0.14,
0.00, 0.00, 0.00, 0.00, 0.00, 0.00,
0.00; 0.00, 0.00, −0.80, 0.00, −0.63,
0.00, −0.52, 0.00, 0.00, −0.45, 0.00,
0.00, −0.41]
0.0 82.27
1 [4, 6, 8, 10, 12, 14, 16, 17, 18, 19, 20,
21, 22; 6, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
3, 3]
82.27 0.00
k = 18 0 [4, 6, 7, 8, 9, 10, 11, 12,13, 14, 15, 16,
17, 18, 19, 20, 21, 22; 5, 3, 3, 3, 3,
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3]
[−1.79, 0.00, 0.00, 0.00, 0.00, 0.00,
0.00, 0.00, 0.00, 0.00, 0.00, 0.00,
0.00, 0.00, 0.00, 0.00, 0.00, 0.00;
−0.68, 0.00, −0.88, 0.00, 0.00,
−0.76, 0.00, 0.00, −0.64, 0.00, 0.00,
−0.54, 0.00, 0.00, −0.46, 0.00, 0.00,
−0.41]
0.0 80.80
1 [4, 6, 7, 8, 9, 10, 11, 12,13, 14, 15, 16,
17, 18, 19, 20, 21, 22; 5, 3, 3, 3, 3,
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3]
80.80 0.00
Note 1: The first three numbers represent the application years of overlays in two digits after year 2000 and the next three numbers the thicknesses
of overlays in centimeters.
Note 2: Convergence index defined in inequality (8).
True optimization of pavement maintenance options 203
Table 8
Exhaustive search for optimal k with only OMW(setting 2: a high volume case)
i (Run NB(Xi) CI (%)
number) Xi (note 1) [ ti ; wi ] ∇NB(Xi) [∂ NB∂ti ;
∂ NB
∂wi
] αi ($ mill.) (note 2)
k = 3 0 [4, 10, 16; 6, 6, 3] [−6.20, −0.33, −0.05; 2.43, 0.98, 1.02] 3.5 215.54
1 [4, 10, 16; 7, 7, 7] 220.38 2.25
k = 4 0 [4, 10, 14, 18; 7, 4, 4, 4] [−6.74, −0.44, −0.12, −0.15; 0.00,
−0.24, −0.05, −0.87]
0.0 221.40
1 [4, 10, 14, 18; 7, 4, 4, 4] 221.40 0.00
k = 5 0 [4, 9, 13, 17, 21; 7, 4, 4, 4, 3] [−7.17, −0.07, −0.04, −0.25, −0.78;
0.00, −0.51, −0.24, −0.10, −0.33]
0.0 221.43
1 [4, 9, 13, 17, 21; 7, 4, 4, 4, 3] 221.43 0.00
k = 6 0 [4, 8, 12, 15, 18, 21; 7, 3, 3, 3, 3, 3] [−7.25, 0.02, −1.70, −1.40, −0.36,
−0.58; 0.00, −0.62, −0.40, −0.30,
−0.26, −0.36]
0.0 222.11
1 [4, 8, 12, 15, 18, 21; 7, 3, 3, 3, 3, 3] 222.11 0.00
k = 7 0 [4, 7, 10, 13, 16, 19, 21; 6, 3, 3, 3, 3,
3, 3]
[−7.60, −0.84, −1.10, −1.67, −0.58,
−0.15, −0.53; −0.30, −0.53,−0.58,
−0.38, −0.19, −0.30, 0.00]
0.0 221.23
1 [4, 7, 10, 13, 16, 19, 21; 6, 3, 3, 3, 3,
3, 3]
221.23 0.00
k = 9 0 [4, 7, 9, 11, 13, 15, 17, 19, 21; 6, 3, 3,
3, 3, 3, 3, 3, 3]
[−7.29, 0.16, 0.28, 0.00, 0.82, −0.24,
0.70, −0.67, −0.27; 0.43 −0.29,
0.00, 0.16, 0.00, 0.38, 0.00, 0.26,
0.00]
0.7 220.66
1 [4, 7, 9, 11, 14, 15, 17, 19, 21; 6, 3, 3,
3, 3, 3, 3, 3, 3]
221.19 0.24
k = 12 0 [4, 6, 8, 10, 12, 14, 16, 18, 19, 20, 21,
22; 6, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3]
[−6.59, 0.55, 0.73, 0.86, −0.33, 0.94,
−0.37, 0.00, 0.00, 0.00, 0.00, 0.00;
0.56, 0.00, 0.06, 0.00, 0.17, 0.00,
0.04, 0.00, −0.14, 0.00, 0.00, −0.39]
0.6 220.81
1 [4, 6, 8, 10, 12, 15, 16, 18, 19, 20, 21,
22; 6, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3]
221.15 0.15
k = 18 0 [4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22; 6, 3, 3, 3,
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3]
[−5.97, 0.00, 0.00, 0.00, 0.00, 0.00,
0.00, 0.00, 0.00, 0.00, 0.00, 0.00,
0.00, 0.00, 0.00, 0.00, 0.00, 0.00;
−0.30, 0.00, −0.75, 0.00, −0.59,
0.00, 0.00, −0.42, 0.00, 0.00, −0.28,
0.00, 0.00, 0.00, −0.24, 0.00, 0.00,
−0.40]
0.0 220.70
1 [4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 7, 18, 19, 20, 21, 22; 6, 3, 3, 3,
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3]
220.70 0.00
Note 1: The first three numbers represent the application years of overlays in two digits after year 2000 and the next three numbers the thicknesses
of overlays in centimeters.
Note 2: Convergence index defined in inequality (8).
from being trapped in local optima. Given no substantial
difference in the performance of the two gradient meth-
ods, the steepest descent method is a recommended algo-
rithm to be used with HDM-4. The studies presented in
this article used HDM-4, and a number of runs were man-
ually made, as it is not available in batch mode. If avail-
able, however, the procedure presented in this article
may easily be coded in a computer program that would
require no exogenously specified maintenance options
but compute optimal option as output.
The proposed methodology can readily be generalized
for cases with maintenance options with heterogeneous
works. Also, it seems to be applicable with other what-
if models to find the true optima, making them more
204 Tsunokawa, Van Hiep & Ul-Islam
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
80 
70 
90 
200 
190
NB(k) 
Setting 2: high vol 
Setting 1: low vol. 
k 
Fig. 6. OMW-optimized net benefit function, NB(k).
powerful tools for analyzing different management
issues.
Although the results of the case study seem to indicate
that the HDM-4 objective function is unimodal with re-
spect to the variables defining maintenance options, and
therefore the solutions obtained are not dependent on
the assumed initial maintenance options, it is not possi-
ble to prove that is always the case. Also, the number of
iterations to find an optimum with the proposed proce-
dure depends on the closeness of the initially assumed
option to the true optimum. These considerations point
to the importance of the development of a method for
finding a good initial option that well approximates the
optimum. The procedure developed by Tsunokawa and
Schofer (1994) based on an optimal control model ap-
pears to be a promising approach for developing such a
method.
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