Comparison of Pavement Network Management Tools based
on Linear and Non-Linear Optimization Methods
Lijun Gao1, Eddie Y. Chou2, and Shuo Wang3
(1) Department of Civil Engineering, University of Toledo,
2801 W. Bancroft Street, Toledo, OH 43606-3390 Phone: (419)530-8058
E-mail: [email protected]
(2) Department of Civil Engineering, University of Toledo,
2801 W. Bancroft Street, Toledo, OH 43606-3390 Phone: (419)530-8123
Fax: (419)530-8116 E-mail: [email protected]
(3) Department of Civil Engineering, University of Toledo,
2801 W. Bancroft Street, Toledo, OH 43606-3390 Phone: (419)530-8058 E-mail: [email protected] Submission Date: 08/01/2012 Word Count: Body Text = 3,465 Abstract = 137 Tables 4 x 250 = 1,000 Figures 5 x 250 = 1,250 Total = 5,852
ABSTRACT
1 2
Transportation officials at the state and local governments must make the best use of available
3
budget to maintain the existing street network under the current budget environment.
4
Mathematical optimization models can help to identify the optimal maintenance and
5
rehabilitation strategy. This paper compares a linear versus a non-linear optimization model for
6
pavement network management. Both models use the Markov transitional probabilities to
7
represent pavement network deterioration, and both have been implemented as spreadsheet tools
8
for use by municipalities. These tools allow users to estimate the minimum budget required to
9
maintain a pavement network to achieve a desired network condition. For a given budget, the
10
tools can be used to determine the budget allocation required to achieve the best network
11
condition. The algorithms for both models are presented, and the results produced from the two
12
models are compared.
INTRODUCTION
1 2
How to best utilize available funds to maintain and improve street network condition is a
3
challenging task facing many local transportation officials. The City of Toledo has experienced
4
deteriorating pavement network condition in recent years, particularly in its residential street
5
network, due to delayed maintenance and reductions in its capital improvement budget. A tool
6
aimed at identifying and optimizing the maintenance and rehabilitation strategy to improve the
7
city‘s pavement condition has been developed. This tool uses the Markov transition probability
8
concept (1) to represent pavement network deterioration, and uses either linear or non-linear
9
programming (1) to model the optimization problems of: (1) finding the minimum budget
10
required to achieve the desired level of pavement network condition, or (2) deciding the best
11
budget allocation between maintenance and rehabilitation to achieve the highest possible
12
network condition for a given budget.
13
The tool is implemented in Microsoft Excel and uses the Excel‘s optimization function,
14
Solver, to solve the optimization problem. With Excel Solver, an optimal (maximum or
15
minimum) value for a formula in one cell — called the objective cell — can be determined,
16
subject to constraints, or limits, on the values of other formula cells on a worksheet. The Solver
17
works with a group of cells, called decision variables, to participate in computing the formulas in
18
the objective and constraint cells. The Solver adjusts the values in the decision variable cells to
19
satisfy the limits on constraint cells and produce the optimal value for the objective cell.‖(2)
20
The Excel Solver uses the Simplex Method to solve linear optimization problems. For
21
smooth non-linear optimization problems, it uses the Generalized Reduced Gradient (GRG)
22
method, and for non-smooth nonlinear optimization problems, it uses the Evolutionary method
23
(3). By integrating the Markov transition probability model with the Solver‘s linear and
non-24
linear optimization solutions, a linear and a non-linear optimization models were formulated in
25
the spreadsheet.
26
In this study, the residential street pavement network condition data from the City of
27
Toledo were analyzed and optimized by both the linear and nonlinear models, and the resulting
28
treatment polices were compared. Both the GRG and Evolutionary methods were used to solve
29
the nonlinear model. The same results were produced, but the Evolutionary method took
30
significantly longer time to solve the problems.
31
LITERATURE REVIEW
32
Linear programming and non-linear programming are the two main types of algorithms utilized
33
by researchers in developing pavement management optimization models. In linear
34
programming models, a key assumption is that all functions including objective and constraint
35
function are linear, while in non-linear programming, this assumption does not hold (1).
36
In previously proposed optimization models, Golabi et al. (4), and Chen et al. (5), utilized
37
linear programming, Abaza and Ashur (6) developed their model based on nonlinear
38
programming. Pavement condition prediction model is an important component of a pavement
39
optimization model. There are two main types of prediction models: deterministic models and
40
probabilistic models. According to Butt et al. (7), the pavement deterioration rates are often
41
―uncertain‖. Therefore, the probabilistic model based on the Markov process is the most
42
frequently used approach (4, 5, 6).
43 44
OBJECTIVE, ASSUMPTIONS AND CONSTRAINTS
1 2
The linear and non-linear models have the same objective, which is to minimize the total repair
3
cost of the pavement network to achieve a desired future condition level. They also shared
4
common assumptions and constraints, although expressed in different formats. The total mileage
5
of the studied pavement network is assumed to remain constant. This assumption is reflected in
6
the Sum-to-One constraints. The pavement network is divided into five pavement condition
7
states (1, Excellent; 2, Good; 3, Fair; 4, Poor; 5, Very Poor). Pavements in each condition state
8
may be treated by one or more of the following five repair treatments: 0), Do Nothing; 1), Crack
9
Sealing; 2), Thin Overlay; 3), Resurfacing; 4), Reconstruction. The non-linear model only
10
allows one single treatment policy for the whole study period in terms of the percentage of the
11
network in each condition state receiving treatment, while the linear model allows a different
12
treatment policy for each different analysis year. Table 1 contains the treatments options eligible
13
for each condition class defined in this study.
14 15
TABLE 1 Allowable Treatments
16 17
Treatment Options
Condition Do Nothing Crack Sealing Thin Overlay Resurfacing Reconstruction
Excellent Yes
Good Yes Yes
Fair Yes Yes
Poor Yes Yes
Very Poor Yes Yes
18
The state transition constraint is the key component in both models. It integrates the
19
network condition deterioration trend to the underlying optimization problem, and determines the
20
pavement condition for the following year.
21
The non-negativity constraints ensure that all variables in the optimization model are
22
non-negative. The target condition constraints are used to specify the desired proportion of
23
pavement in each condition state. For example, pavements in Poor and Very Poor conditions are
24
considered as deficient, so certain limits, say, no more than 10% and 5% respectively, may be set
25
as the target condition constraints.
26
If the desired target condition of the pavement network is better than the existing
27
condition, both the linear and non-linear models tend to demand a large amount of treatments to
28
achieve the desired condition level in the beginning of the analysis period. Therefore, the initial
29
several years would require very high, sometimes unrealistic, budgets. To prevent this, the
30
budget constraints can be used to ensure that the required budgets recommended by the model
31
will not exceed the maximum possible budget each year, thus providing a more balanced
32
treatments distribution over the entire study time period, and smooth out the initial high budget
33
demands.
34 35
LINEAR MODEL ALGORITHM
1 2
The linear model is set up in such a way that each analysis year may have a different treatment
3
policy, which is defined as the percentages of pavement network receiving various treatments,
4
including the Do Nothing treatment.
5
In the objective function equation (1), Ytk'ik is the decision variable representing the
6
proportion of pavement in condition state i with last treatment k’ receiving recommended repair
7
treatment k in year t.
8
The state transition constraints equation (2) is applied in the model starting at the second
9
analysis year, the proportion of pavement in condition state j with last treatment k’ in year t is
10
derived from two parts of pavements in different condition states in year t-1: one part with last
11
treatment k’ received no new treatment and the other part received new treatment k’.
12
Equation (3) is the non-negative constraints. Equation (4) is the sum-to-one constraint
13
that reflects the assumption that the total network size remains constant.
14
The target condition constraints equation (5) ensures the proportion of pavement in
15
certain condition states is in a defined range for the final year.
16
The budget constraints equation (6) is used to ensure that the required budgets
17
recommended by the optimized solution do not exceed the maximum available budget each year.
18
Based on the above discussion, the linear model for pavement maintenance and
19
rehabilitation optimization is formulated as follows:
20 21 Minimize 22
T t K k i K k ik tk ik tk C Y 1 ' 1 5 1 0 ' ' (1) 23 24 Subject to 25 26State transition constraints:
27
5 1 ' 0 ' ) 1 ( 0 5 1 1 ' ' ) 1 ( ' i ij k i k t K k i K k ij k kik t jk tk Y P Y DN Y 28 for all t = 2,…, T; k’ = 1,…, K; j = 1,…, I; (2) 29 30 Non-negativity constraints: 31 0 'ik tk Y for all t = 1,…, T; k’ = 1,…, K; i = 1,…, I; k = 0,…, K (3) 32 33 Sum-to-one constraints: 34 1 1 ' 5 1 0 '
K k i K k ik tk Y for all t = 1,…, T (4) 35 36Target condition constraints:
37 Tj Tj S for selected j (5) 38 39 Budget constraints: 40
t T t K k i K k ik tk ik tk C L B Y
1 ' 1 5 1 0 ' ' for all t = 1,…, T (6) 1 2 Where 3 ik tkY ' proportion of pavement in state i with last treatment k’ receiving new
4 treatment k in year t. 5 ik tk
C ' unit cost of applying treatment k in year t to pavement in state i with last
6
treatment k’.
7
kij
P probability that pavement receiving new treatment k transit from state i to
8 state j. 9 ij k
DN ' probability that pavement with last treatment k’ receiving no new
10
treatment (Do Nothing) moves from state i to state j
11
Tj
S Proportion of pavement in sate j at year at final year T
12
it
upper limit of proportion of pavement in condition i in year t.
13
L total length of entire pavement network
14
t
B maximum available budget in year t.
15
T number of analysis years
16
K number of repair treatment types
17 18
NON-LINEAR MODEL ALGORITHM
19 20
In the non-linear model, all years in the analysis period have the same treatment policy. In other
21
words, the same percentages of the pavement network will receive a particular treatment every
22
year. This offers the maintenance agency a simple and consistent treatments policy for
23
implementation. In the objective function equation (7), Xj is the proportion of pavement in
24
state j receiving the yearly treatments.
25
The state transition constraints equation (8) is applied in the model starting at the second
26
analysis year, the proportion of pavement in condition state j in year t is derived from two parts
27
of pavements in different condition states ( i=1,2,…5 ) in year t-1: one part received no treatment
28
and the other part received the yearly treatment improvement.
29
Equation (9) is the non-negativity constraints. Equation (10) represents the sum-to-one
30
constraints, which is used to ensure the overall network size remains constant, by adding the
31
proportion of the pavement received no treatment and the proportion received a new treatment in
32
all condition states. Equation (11) is the target condition constraints. Equation (12) is the budget
33
constraints.
34
Based on the above discussion, the non-linear programming model for pavement
35
maintenance and rehabilitation optimization is formulated as follows:
36 37 Minimize 38
T t j j j tjX LC S 1 5 1 (7) 39 40Subject to
1 2
State transition constraints:
3 (8) 4 for all t = 2,…, T; j = 1, 2,… 5; 5 6 Non-negativity constraints: 7 0 i
X for all i=1,…5 (9)
8 9
Sum to one constraints:
10 1 4 0
k ikX for all i=1,…5 (10)
11 12
Target condition constraints:
13 Tj Tj S for selected j (11) 14 15 Budget constraints: 16 t j j j tjX L C B S
5 1 for t = 1,…t (12) 17 18 Where 19 tjS Proportion of pavement in sate j at year t
20
i
X Proportion of pavement in state i receiving treatment
21
T Number of analysis years
22
L Total length of entire pavement network
23
j
C Unit cost of applying treatment to pavement in state j
24
ij
DN Probability that pavement receiving no treatment (Do Nothing) moves from
25
state i to state j
26
ij
P Probability that pavement receiving new treatment transit from state i to state j
27
Tj
Upper limit of proportion of pavement in condition j in final year T
28
t
B Maximum available budget in year t.
29 30 31
LINEAR AND NON-LINEAR OPTIMIZATION COMPARISIONS
32 33
Two case studies are conducted to compare the minimum budget required by the two models to
34
manage a pavement network. The City of Toledo residential street network is used as an
35
example.
36 37 38
Case I
1 2
Model Input and Constraints
3 4
The required input includes: (1) existing network condition distribution, (2) analysis period in
5
years, (3) target network condition, and (4) unit cost of different repair treatments. The
6
pavement deterioration trend is an optional input as a default trend is provided. Additional
7
constraints such as annual budget limits can be added.
8
The residential street network at the City of Toledo has a total of 744.98 lane miles. The
9
network condition can be described by the percentages of network in the following five condition
10
states: Excellent, Good, Fair, Poor, and Very Poor. Table 2 shows the estimated residential street
11
network condition at 2013. The analysis period is 10 years, starting from year 2013.
12
The Target network condition is the desired network condition, which can be specified by
13
the user in terms of the percentages of network in each of the condition states. In this case study,
14
the Target is to limit the amount of very poor pavements to its current level of 18%.
15
The unit costs per lane mile for each repair treatment option used in the analysis are:
16
Crack Sealing: $2,500; Thin Overlay: $100,000; Resurfacing: $200,000; and Reconstruction:
17
$600,000.
18
Table 3 shows the Markov transitional probability matrix which represents how a typical
19
pavement deteriorates naturally without any repair. In Figure 1, the bottom curve that
20
deteriorates from a score of 95 to 55 in 35 years illustrates this deterioration trend.
21
Table 4 shows the assumed probability that a pavement will be in various condition states
22
after different repair treatments. In Figure 1, the corresponding effectiveness of various repair
23
treatment options are shown.
24 25
TABLE 2 Projected 2013 Pavement Condition
26
Excellent Good Fair Poor Very Poor
Lane-miles 17.06 144.62 345.97 101.92 135.40
Percentage 2% 19% 46% 14% 18%
27
TABLE 3 Markov Transition Matrix (Do Nothing)
28
Excellent Good Fair Poor Very Poor
Excellent 81% 19% 0% 0% 0% Good 0% 85% 15% 0% 0% Fair 0% 0% 90% 10% 0% Poor 0% 0% 0% 67% 33% Very Poor 0% 0% 0% 0% 100% 29
TABLE 4 Post Repair Condition Distribution
30
Excellent Good Fair Poor Very Poor
Crack Sealing 0% 95% 5% 0% 0%
Thin Overlay 30% 70% 0% 0% 0%
Resurfacing 60% 40% 0% 0% 0%
1
2
FIGURE 1 Deterioration trend.
3 4
The objective is to find the minimum budget required to limit the percentage of pavement
5
network in ‗very poor‘ condition state to 18%, under two scenarios: first, without annual budget
6
limit, and second, with a maximum annual budget of $15 million.
7 8
Model Outputs:
9 10
With the aforementioned inputs and constraints, the linear and non-linear model each
11
generated one set of optimized treatment policy and resulted pavement condition without budget
12
constraint. Figure 2 (a) shows budget comparison between linear and non-linear methods.
13
Figure 2 (b) and Figure 2 (c) show the similar pavement conditions projected by the two models
14
respectively, as results of similar treatment policy.
15
It can be seen that the two models produced nearly identical results under no annual
16
budget limit scenario. Both require a total of about $75 million for the 10 year-period, with a
17
significant portion, over $20 million needed in the initial year.
18
Since such a heavily front-loaded budget requirement is not likely practical, a maximum
19
annual budget limit of $15 million is added as a constraint.
1 (a) 2 3 4 (b) 5 6 $0 $5 $10 $15 $20 $25 2013 2015 2017 2019 2021 2023 R e q u ir e d B u d ge t ($ M ill io n s) Analysis Year linear Method Non Linear Method
0%
20%
40%
60%
80%
100%
1
(c)
2
FIGURE 2 Case I comparison results without budget constraint: (a) Case I budget
3
comparison without budget constraint, (b) Case I linear model projected pavement
4
conditions without budget constraint, (c) Case I non-linear model projected pavement
5
conditions without budget constraint.
6 7
With annual budget constraint of $15 million, to limit the percentage of very poor
8
pavement to 18%, the two models project different budget requirements. The linear model
9
results show a total of about $79 million is required for the 10 year-period, or on average, $7.9
10
million per year. The solution of the non-linear model requires a total of about $108 million for
11
the 10 year-period, or $10.8 million per year. That is 37% more than linear method. Figure 3
12
(a) shows the budget comparison between the two. Figure 3 (b) and Figure 3 (c) show the
13
pavement conditions projected by the linear and non-linear models respectively.
14
Results from both models again show that more budgets are required in the initial year(s),
15
because the projected network conditions in later years are based on the network condition in
16
initial year through the Markov transition matrix. The total cost to maintain the network for the
17
entire analysis period would be lower when the network is in better condition to begin with.
18
Therefore, the solutions from the optimization models recommend spending as much budget as
19
possible in the beginning of the analysis period to bring the network to a very good condition
20
level, and save money in later years‘ maintenance and repair costs. This lowers the overall cost
21
in the long run, but the initially high cost and uneven budget requirements may not be practical
22
in most government agencies.
23
The results of the linear and non-linear models also differ in one important aspect. That
24
is, the linear model produces a different treatment strategy for each year during the analysis
25
period, while the non-linear model generates one treatment strategy for the entire analysis period.
26
A treatment strategy is expressed as the proportions of pavements in various condition states to
27
receive certain treatment. For example, assuming pavements in Very Poor condition requires
28
reconstruction, not all Very Poor pavements can be reconstructed due to the high cost may
29
exceed the available budget. Instead, only a portion of the Very Poor pavements may be
30
reconstructed. Because the linear model can vary treatment strategy from one year to the next, it
31
is more flexible and efficient, and hence results in significant less overall budget. The drawback
32
of this flexibility is more significant fluctuation of the required budget from year to year.
33
0%
20%
40%
60%
80%
100%
1 2 (a) 3 4 5 6 7 (b) 8 9 $0 $2 $4 $6 $8 $10 $12 $14 $16 2013 2015 2017 2019 2021 2023 R e q u ir e d B u d ge t ( $ M ill io n s) Analysis Year
linear Method Non Linear Method
0%
20%
40%
60%
80%
100%
1 2
(c)
3 4
FIGURE 3 Case I comparison results with budget constraint: (a) Case I budget comparison
5
with budget constraint, (b) Case I linear model projected pavement conditions with budget
6
constraint, (c) Case I non-linear model projected pavement conditions with budget
7 constraint. 8 9 Case II 10 11
Model inputs and constraints:
12 13
In this case, the objective is to find the minimum budget required to reduce the percentage of
14
―very poor‖ pavements in the network from 18% to 10% over the 10 year time period, under two
15
scenarios: first, without annual budget limit, and second, with a maximum annual budget of $20
16
million. All the other inputs and constraints remain the same as in Case I.
17 18
Model Outputs:
19 20
Figure 4 (a) shows the required budget comparison between the linear and non-linear
21
models. Even though the required budgets produced by the linear model vary far more
22
significantly during the analysis period, both models require a total of about $111 million for the
23
10 year-period, or $11.1 million per year, on average. Figure 4 (b) and Figure 4 (c) show the
24
network pavement conditions projected by the two models are very similar.
25 26 27
0%
20%
40%
60%
80%
100%
1 2 (a) 3 4 5 6 (b) 7 8 $0 $5 $10 $15 $20 $25 $30 2013 2015 2017 2019 2021 2023 R e q u ir e d B u d ge t ( $ M ill io n s) Analysis Year
linear Method Non Linear Method
0%
20%
40%
60%
80%
100%
1
(c)
2 3
FIGURE 4 Case II comparison results without budget constraint: (a) Case II budget
4
comparison without budget constraint, (b) Case II linear model projected pavement
5
conditions without budget constraint, (c) Case II non-linear model projected pavement
6
conditions without budget constraint.
7 8 9
In the second scenario, a maximum annual budget limit of $20 million is added as a
10
constraint. The resulting linear model projects a total of about $112 million for the 10
year-11
period, or $11.2 million per year, which is only slightly higher than no annual budget limit
12
scenario. Figure 5 (a) shows the linear method budget requirement trend under this scenario.
13
Figure 5 (b) shows the pavement conditions projected by the linear model.
14
The non-linear model cannot find a solution under this scenario, because the non-linear
15
model lacks the flexibility of the linear model in varying the proportion of network receiving
16
different treatment each year, i.e. non-linear model offers only one set of treatment policy for all
17
the analysis year in terms of the percentage of the network in each condition state receiving
18 treatment. 19
0%
20%
40%
60%
80%
100%
1 (a) 2 3 4 (b) 5 6
FIGURE 5 Case II results with budget constraint: (a) Case II linear model budget trend
7
with budget constraint, (b) Case II linear model projected pavement conditions with
8 budget constraint. 9 10 11 12 $0 $5 $10 $15 $20 $25 2013 2015 2017 2019 2021 2023 R e q u ir e d B u d ge t ( $ M ill io n s) Analysis Year linear Method
0%
20%
40%
60%
80%
100%
CONCLUSIONS
1 2
The spreadsheet optimization tool described in this paper can be used by local
3
transportation officials to estimate the budget requirement to maintain or improve their pavement
4
networks. From the analysis and comparison of the two Cases with and without annual budget
5
constraint, it can be seen that without annual budget constraint, the two models generated nearly
6
identical results: almost the same total budget requirement, similar treatment plans and projected
7
pavement conditions. However, when the annual budget constraint is added, the linear model
8
produces a lower required budget than the non-linear model. If the annual budget constraint
9
added is somewhat stringent, the Solver may not be able to find a solution for the non-linear
10
model. This is due to the non-linear model offers only one set of treatment policy for all the
11
analysis year in terms of the percentage of the network in each condition state receiving
12
treatment, while the linear model has the flexibility in varying the proportion of network
13
receiving different treatment each year. Based on the case study results, the non-linear model
14
appears to produce a more conservative solution and requiring more budgets on average.
15
However, the non-linear model produces one consistent repair treatment policy which may be
16
easier for agencies to adopt.
17 18
1
REFERENCES
2 3
1. Hillier, F. S. and G. J. Lieberman. Introduction to Operations Research, 9th ed.,
McGraw-4
Hill, 2010.
5 6
2. 2010 Excel Help: Analyzing data/ What-if analysis/ Define and Solve a problem by using
7
Solver.
8 9
3. 2010 Excel Help: Analyzing data/ What-if analysis/ Define and Solve a problem by using
10
Solver/Solving methods used by Solver
11 12
4. Golabi, K., R. B. Kulkarni, and G. B. Way. A Statewide Pavement Management System.
13
Interfaces, Vol. 12, No. 6, 1982, pp. 5-21.
14 15
5. Chen, X., S. Hudson, M. Pajoh, and W. Dickinson. Development of New Network
16
Optimization Model for Oklahoma Department of Transportation. In Transportation
17
Research Record: Journal of the Transportation Research Board, No. 1524, Transportation
18
Research Board of the National Academies, Washington, D.C., 1996, pp. 103-108.
19 20
6. Abaza, K. A., and S. Ashur, Optimum decision policy for management of pavement
21
maintenance and rehabilitation, In Transportation Research Record: Journal of the
22
Transportation Research Board, No. 1655, Transportation Research Board of the National
23
Academies, Washington, D.C., 1999, pp. 8–15.
24 25
7. Butt, A. A., M. Y. Shahin, S. H. Carpenter, and J. V. Carnahan. Application of Markov
26
Process to Pavement Management Systems at Network Level. Proceedings of 3rd
27
International Conference on Managing Pavements, San Antonio, Texas, May 1994.
28 29
