NetworkLevel Road Pavement Maintenance ... - Wiley Online Library

Computer-Aided Civil and Infrastructure Engineering 27 (2012) 276–287

Network-Level Road Pavement Maintenance and Rehabilitation Scheduling for Optimal Performance Improvement and Budget Utilization Lu Gao, Chi Xie & Zhanmin Zhang Department of Civil, Architectural and Environmental Engineering, Center for Transportation Research, The University of Texas at Austin, TX 78701, USA

& S. Travis Waller∗ School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW 2052, Australia

Abstract: This article discusses how to efficiently and completely solve a bi-objective pavement maintenance and rehabilitation-scheduling problem, which aims at optimizing two objectives of pavement condition improvement and budget utilization in a simultaneous manner. This problem may be addressed by the weighting method, constraint method, ranking method, and various metaheuristic methods. However, none of these methods can guarantee the complete Pareto-optimal solution set, which would potentially lead to suboptimal decisions. In this article, a parametric method is suggested to solve the bi-objective pavement maintenance and rehabilitation-scheduling problem. The effectiveness and efficiency of the parametric method is investigated and demonstrated through a case study using the real-world data set from the Dallas District’s Pavement Management Information System. A performance comparison between the widely used weighting method and the parametric method clearly justifies the computational advantages of the parametric method.

∗ To

whom correspondence should be addressed. E-mail: s.waller@ unsw.edu.au.

C 2011 Computer-Aided Civil and Infrastructure Engineering. DOI: 10.1111/j.1467-8667.2011.00733.x

1 INTRODUCTION Pavement maintenance over a road network consists of various routine, preventive, or reactive activities, including filling cracks, patching potholes, and other applicable techniques, such as chip seal coating or use of a slurry seal. Pavement rehabilitation, on the other hand, involves actions to enhance the structural capacity of pavements, such as resurfacing (overlay), resurfacing with partial reconstruction (localized reconstruction), and complete reconstruction. In combination, road pavement maintenance and rehabilitation (M&R) becomes one of the costly transportation infrastructure activities (Wang et al., 2003; Abbas et al., 2007; Byrne et al., 2009; Yang et al., 2009; Ying and Salari, 2010; Deshpande et al., 2010; Bianchini and Bandini, 2010; Lajnef et al., 2011; Wang and Li, 2011). In scheduling M&R activities for an urban or rural transportation network, decision makers face great challenges of determining which pavement sections are to be repaired, when and how repairs should be carried out, and what treatment to use. In general, there are two types of M&R scheduling problems often encountered by pavement managers and planners, namely, the budget planning problem and the budget allocation problem. A budget-planning problem aims to minimize the total M&R cost over a

Network-level road pavement maintenance and rehabilitation scheduling

certain planning horizon, such that a specific condition requirement is satisfied at the network level or at the individual pavement section level. The budget-planning problem is usually solved before the amount of the budget is known. The solved minimum expenditure is then used to help higher-level decision makers determine the actual budget for the planning horizon. On the other hand, a budget allocation problem tries to maximize the M&R effectiveness or minimize the user cost, subject to some budget constraints. The budget allocation problem is often solved after the limit of the available budget is known to the pavement management agency (Haas et al., 1994). Although the budget planning and allocation problems are typically treated and solved separately (see Friesz and Fernandez, 1979; Guignier and Madanat, 1999; Ouyang and Madanat, 2004; Deshpande et al., 2010; and Gao et al., 2010, for example), the decision maker will obtain a richer body of information and make a more comprehensive evaluation if both problems can be solved simultaneously as a bi-objective optimization problem. Although these single-objective problems are formed by inserting one objective into the objective function with folding the other objective as the problem’s constraint, the alternative way we describe here is to formulate a bi-objective problem, which is formed by putting both the objectives into a vector function. Choosing the single-objective or bi-objective approach really depends on the user’s actual need. In many cases, if the budget limit cannot be firmly determined a priori or its value could vary in a range, it is natural that we use a bi-objective approach to simultaneously optimize the two objectives and then make the final decision in terms of the solution distribution of the resulting Pareto-optimal set as well as other external constraints or considerations. Several modeling and solution efforts have been made by researchers to address the bi-objective M&R scheduling problem. For example, Liu et al. (1997) proposed using genetic algorithm (GA) to solve a biobjective optimization model for bridge deck rehabilitation. Two objectives are defined as the minimization of the rehabilitation cost and the maximization of the deterioration degree. Similar GA-based approaches can also be found in the work of Liu and Frangopol (2005), Neves et al. (2006), and Fwa et al. (2000) for pavement management problems, and Adeli and Cheng (1994a, b), Hung and Adeli (1994), Adeli and Kumar (1995a, b), Sarma and Adeli (1998), Kim and Adeli (2001), Vlahogianni et al. (2007), Jiang and Adeli (2008), Cheng and Yan (2009), Kang et al. (2009), Lee and Wei (2010), Al-Bazi and Dawood (2010), Marano et al. (2011), and Baraldi et al. (2011) for other

277

civil infrastructure management problems. The disadvantage of the GA approach, due to its heuristic nature, is that there is no guarantee for finding optimal solutions in a finite amount of time and the parameter tuning is extremely time-consuming. Wang et al. (2003) proposed a bi-objective integer programming (IP) model to formulate the M&R scheduling problem. One objective of this model is to maximize the effectiveness of pavement maintenance treatments while the other is to minimize the maintenance-incurred user disturbance cost. Exact solution methods for such a bi-objective IP model, however, only work tractably for small-sized and homogenous network-level management problems. The problem’s computing cost increases exponentially as the size of the network increases. Wu and Flintsch (2009) resorted to a bi-objective linear programming (LP) model for formulating the M&R scheduling problem and solving it by the weighting method. In the LP approach, pavement sections with similar characteristics are grouped together. In this regard, the computational effort for solving this type of problems is much lower than their IP counterparts. However, the weighting method in general does not guarantee the complete set of Pareto-optimal solutions. In this article, we present an efficient solution method, namely parametric method, for the aforementioned bi-objective LP-based M&R scheduling problem. This problem has been addressed by previous researchers using the weighting method, constraint method, ranking method, and various metaheuristic methods. However, none of these methods can guarantee the complete Pareto-optimal solution set, which would potentially lead to suboptimal decisions (see Xie and Waller, 2011). For a bi-objective optimization problem of linear structure, the parametric method is superior in both solution quality and efficiency to the widely used weighting method. Given a pre-specified weight factor set, the weighting method may omit some Pareto-optimal solutions in solution-dense feasible regions and perform some resultless searches in solutionsparse regions. Due to this disadvantage, the weighting method may only be regarded as heuristics for bi-objective or multiobjective problems. The parametric method avoids this incompleteness and inefficiency mentioned above by systematically identifying those “critical” weighting parameter sets so that all Paretooptimal basic feasible solutions and hence the whole Pareto-optimal frontier can be exhausted by repeatedly solving a series of single-objective LP problems. The remaining part of the article is arranged in four sections. First, we discuss the motivation and formulation of the single-objective and bi-objective M&R scheduling problems in the pavement management

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Gao, Xie, Zhang & Waller

context. Then, we describe the parametric method and detail how this method is adapted to the given biobjective M&R scheduling problem. An example application of the model and solution method for a real-world case study is then presented to illustrate the effectiveness and efficiency of the solution method. In particular, the computational performance of the parametric method is highlighted by a comparison of this method and the well-known weighting method both implemented in the example problem. The last section concludes the article by highlighting the advantages of the parametric method for the bi-objective M&R scheduling problem and figures out some other types of M&R scheduling problems to which the parametric method is potentially applicable.

2 PROBLEM FORMULATION Following Gao et al. (2010), the mathematical formulation of a bi-objective pavement M&R scheduling problem for a road network consisting of multiple road groups with similar characteristics is presented in this section. This bi-objective problem can be regarded as a synthetic result of two single-objective M&R scheduling problems, including the budget planning problem and the budget allocation problem. Thus, our discussion in this section gets started with presenting the two singleobjective problems. 2.1 Notation For discussion convenience, we first list the sets, parameters, and variables used in the model and algorithm development in Table 1. 2.2 Pavement M&R scheduling problems 2.2.1 The budget allocation problem (P1). The budget allocation problem (P1) is formulated as in Equations (1)–(5). The objective of P1 is to maximize the proportion of roads in the first (best) condition state over the scheduling period T. This formulation is to accommodate the current Texas practice used in the case study. It can be easily replaced by other indicators of the performance, for example, the average of the whole network, without affecting the solution procedure. Equation (2) represents the initial condition of each road group at the beginning of the scheduling horizon. Equation (3) represents the deterioration process of road conditions between consecutive years. Equation (4) assures that the money spent on M&R activities each year cannot exceed the annual budget. In this article, we assume that the resource cannot be rolled over to the next plan year.

Table 1 Notation Sets S

Set of road groups with similar characteristics and S = {1, 2, . . . , S} Set of pavement condition states and I = {1, 2, . . . , I} with I representing the worst condition state Set of M&R treatments and M = {1, 2, . . . , M} with the Mth treatment being the most effective and expensive

I

M

Parameters Bt Csmt

The available budget limit at year t The unit cost of applying the mth treatment to the sth road group at year t ($/km) Total length of the sth road group Deterioration transition probability from condition state i to state j when the mth treatment is applied to the sth road group. Psi jm satisfies the constraint of Psi jm = 1, s ∈ S, (i, j) ∈ I, m ∈ M

Ls Psijm

j∈I

Xsi1

Proportion of the sth road group in condition state i at the beginning of the first year, which is known to the decision maker before the M&R scheduling Minimum requirement on the proportion of road network in the first condition state

X∗

Decision variables Xsimt

Proportion of the sth road group in condition state i that receives the mth treatment at year t

The value of Bt may be chosen to reflect the time value of money. Equation (5) defines the feasible range for the decision variables. The decision variables of this optimization problem determine how much proportion of each road group will receive an M&R treatment, when treatment should be carried out, and what treatment to use. T S M 1 1 Ls Xs1mt max z = Ls T + 1 t=1 s=1 m=1

s∈S

+

S I M

Psi1m Ls XsimT

(1)

Xsim1 = Xsi1 , ∀s ∈ S, i ∈ I

(2)

s=1 i=1 m=1

subject to :

M m=1


M

Xsjmt =

m=1

M I

Psijm Xsim,t−1 , ∀s ∈ S, j ∈ I,

m=1 i=1

t = 2, . . . , T S I M

Csmt Xsimt Ls ≤ Bt , ∀t ∈ T

(3) (4)

s=1 i=1 m=1

0 ≤ Xsimt ≤ 1, ∀s ∈ S, i ∈ I, m ∈ M, t ∈ T

(5)

2.2.2 The budget planning problem (P2). The budget planning problem (P2) is formulated as in Equations (6)–(11). The objective of P2 is to minimize the average annual M&R cost. Equations (7), (8), and (11) are the same as Equations (2), (3), and (5). Equations (9) and (10) assure that for each year the proportion of road network in the first condition state should be larger than a predefined requirement. min z =

T S I M 1 Csmt Xsimt Ls T

(6)

together, a bi-objective pavement M&R scheduling problem (P3) can be formulated as in Equations (12)–(18). z min z = 1 z2 T S M ⎤ ⎡ 1 1 Ls Xs1mt ⎥ ⎢− ⎥ ⎢ L T+1 t=1 s=1 m=1 ⎥ ⎢ s∈S s ⎥ ⎢ ⎥ ⎢ S I M ⎥ ⎢ ⎥ ⎢ =⎢ + Psi1m Ls Xsimt ⎥ ⎥ ⎢ s=1 i=1 m=1 ⎥ ⎢ ⎥ ⎢ T S I M ⎥ ⎢1 ⎦ ⎣ Csmt Xsimt Ls T t=1 s=1 i=1 m=1

(12)

subject to : M

Xs jmt =

m=1 M

Xsim1 = Xsi1 , ∀s ∈ S, i ∈ I

Xs jmt =

m=1

M I

M I

(13)

Psi jm Xsim,t−1 , ∀s ∈ S, j ∈ I,

m=1 i=1

t = 2, . . . , T S I M

Psi jm Xsim,t−1 , ∀s ∈ S, j ∈ I,

Csmt Xsimt Ls ≤ Bt , ∀t ∈ T

(14)

(15)

s=1 i=1 m=1

m=1 i=1

t = 2, . . . , T

Xsim1 = Xsi1 , ∀s ∈ S, i ∈ I

(7)

m=1 M

M m=1

t=1 s=1 i=1 m=1

subject to :

279

1 S Ls

S M

Xs1mt Ls ≥ X∗ , ∀t = 2, 3, . . . , T

(8) (9)

1 Xs1mt Ls ≥ X∗ , ∀t = 2, 3, . . . , T S Ls s=1 m=1 S

1 S

Ls

1 Psi1m Ls Xsimt ≥ X∗ S Ls s=1 i=1 m=1 S

s=1

Psi1m Ls Xsimt ≥ X∗

(10)

s=1 i=1 m=1

0 ≤ Xsimt ≤ 1, ∀s ∈ S, i ∈ I, m ∈ M, t ∈ T

I

M

(11)

2.2.3 The bi-objective M&R scheduling problem (P3). Solving either of the single-objective problems (P1 and P2) typically results in a single optimal solution that does not consider other system goals. If both the problems are combined and solved simultaneously, the decision maker may obtain more comprehensive information and then make a better trade-off in final decision making in view of all kinds of (varying) constraints and resource limits. By combing P1 and P2

(17)

s=1

0 ≤ Xsimt ≤ 1, ∀s ∈ S, i ∈ I, m ∈ M, t ∈ T

s=1

(16)

s=1

s=1 m=1

S I M

M

(18)

This bi-objective M&R scheduling problem is the target problem we will solve in this article by adapting an efficient solution method.

3 SOLUTION METHOD 3.1 Pareto-optimal solutions To a bi-objective optimization problem, a solution is Pareto-optimal if the improvement in one objective can only be achieved at the expense of disimproving the other objective. Pareto-optimal solutions are also called

280


z2

solutions are obtained sequentially through exhausting the parameter range and solving the corresponding parameterized problems. In this fashion, the bi-objective M&R scheduling problem presented in this article can be reduced to the following combined-objective problem (P4): T S M 1 1 Ls Xs1mt min z = −w1 Ls T + 1

Dominated Solutions

t=1 s=1 m=1

s∈S

+ Nondominated Solutions (Pareto-Optimal Frontier)

S I M

Pi1m Ls Xsimt

s=1 i=1 m=1

+ w2

T S I M 1 Csmt Xsimt Ls T t=1 s=1 i=1 m=1

(19)

-z1

Fig. 1. Illustration of a pareto-optimal frontier.

nondominated solutions because there are no other solutions that are superior to them in both objectives. The Pareto-optimal frontier is defined as the set of all nondominated solutions. Using P3 as an example (see Figure 1), the Pareto-optimal frontier of a bi-objective LP problem has two objectives: maximization of −z1 and minimization of z2 . The gray area represents the whole dominated solution region. The frontier is a piecewise line connecting the nondominated solution points. As seen in Figure 1, for any nondominated solution on the frontier, there is no other feasible solution that is better than it in both objectives. 3.2 The parametric algorithm This section presents the detailed procedure of the parametric method for the proposed bi-objective M&R scheduling problem. The parametric algorithm has its root from a noninferior set estimation method (Cohon, 1978) and guarantees the completeness of the Paretooptimal solution set of a bi-objective LP problem in a limited time frame. The core idea of this method is to reduce an optimization problem with two objectives to a number of single-objective problems through convex combinations with different weighting parameters. The method typically uses a parameterized utility function combining all the objectives so as to convert a multiobjective optimization problem into a series of single-objective problems with a range of parameter values. Each parameter in the utility function serves as a weight for its corresponding objective; Pareto-optimal

subject to constraints (13)–(18). Through exhausting the whole range of w1 and w2 , the whole Pareto-optimal (or nondominated) solutions can be found by solving P4 repeatedly. Geometrically, the parametric method may be depicted as an iterative divide-and-conquer process, which gradually locates a Pareto-optimal basic feasible solution in each divided parameter range confined by two previous Pareto-optimal solution points that are used to generate the current weighting parameter set or concludes no Pareto-optimal solution is found in this range. The algorithmic procedure of the parametric method can be described as follows. Step 0 (Initialization): Give the maximum number of iterations kmax . Set the initial weighting parameters w1 = 1 − ε and w2 = ε, where ε is a sufficiently small number, that is, 0 < ε = 1, and solve the combinedobjective problem (P4) with w = (w1 , w2 ). Label the obtained optimal objective vector as z1 = (z11 , z21 ), where z11 represents the first objective value and z21 the second objective value. Similarly, set w1 = ε and w2 = 1 − ε, and solve the corresponding combined-objective (P4) again and obtain its optimal objective vector z2 = (z12 , z22 ). It is apparent that the two initial parameter sets are so set as to obtain two Pareto-optimal solutions (x1 , x2 ) that are exclusively optimal to the first and second objectives, respectively. Moreover, create a first-in-first-out list to store neighboring Pareto-optimal solution pairs and add (x1 , x2 ) into this list. Set k = 1 and go to Step 1. Step 1 (Parameter generation): Given a pair of neighboring Pareto-optimal solutions, x1 and x2 , as well as their objective vectors z1 = (z11 , z21 ) and z2 = (z12 , z22 ), generate a new weighting parameter set w = (w1 , w2 )


by solving the following simple linear system: ⎡

z11 − z12 z21 − z22

⎤

⎥ w1 ⎢ 2 ⎣ z1 − z11 z22 − z21 ⎦ w2 1 1

Start

⎡ ⎤ 0 w1 ⎢ ⎥ ⎢ ⎥ 0 =⎣ ⎦⇒ w2 1 a1 ⎤ ⎢a +a ⎥ = ⎣ 1a 2 ⎦ 2 a1 + a2 ⎡

281

(20)

where a1 = z22 − z21 and a2 = z11 − z12 . Remove the solution pair (x1 , x2 ) from the neighboring solution pair list. Note that this method of generating weighting parameters is the so-called perpendicular method, which has been used in a number of multiobjective optimization problems, including, for example, Cohon (1978) for a general multiobjective LP problem, Aneja and Nair (1979) for a multiobjective transportation problem, Fruhwirth et al. (1989) for a multiobjective minimum cost flow problem, and Lin and Xie (2009) for a multiobjective network design problem. Other parameter generating methods could be used in our case as well (see Fruhwirth et al., 1989 for alternative methods). Step 2 (Solution generation and examination): Solve the combined-objective problem (P4) with the new weighting parameter set w in Step 1. If the objective vector z of the obtained optimal solution x is identical to either x1 or x2 , conclude that there is no Pareto-optimal basic feasible solution in the feasible objective region: [z12 , z11 ] and [z21 , z22 ], if z11 > z12 and z21 < z22 (or [z11 , z12 ] and [z22 , z21 ], if z11 < z12 and z21 > z22 ). Otherwise, the newly obtained solution x is a Pareto-optimal solution; create two new neighboring Pareto-optimal solution pairs: (x1 , x) and (x, x2 ), and add them into the neighboring solution pair list. Set k = k+1. Step 3 (Stopping criterion check): If k > kmax or the neighboring solution pair list is empty, stop the search. Otherwise, go to Step 1. The complete Paretooptimal piecewise curve can be obtained by connecting a straight line between each pair of neighboring solution points, if the algorithm stops after the neighboring solution pair list is empty. In this case, all Pareto-optimal basic feasible solutions to the problem are found by the algorithm. It is evident that the major computational effort of the above algorithmic procedure is spent on repeatedly solving the combined-objective problem, which is a single-objective LP problem. In our case, we utilize the simplex method for solution of the combined-objective problem. In summary, the above algorithmic procedure is depicted in the following flowchart (see Figure 2).

Step 0: Initialization Set k max and initial values of w1 and w 2 Solve the combined -objective problem given w 1 and w 2 Set k := 1

Step 1: Parameter Generation Calculate w1 and w2, based on x 1 and x2 from the FIFO list, using the perpendicular method

Step 2: Solution Generation and Examination Obtain the optimal solution x by solving the combined -objective problem given w 1 and w 2 If x = x1 or x = x 2, go to the next step; if not, 1 2 store solution pairs (x , x) and (x, x ) into the FIFO list and set k := k + 1

No

Step 3: Stopping Criterion Check Check k > kmax or the solution pair list is empty Yes

End

Fig. 2. The algorithmic procedure of the parametric method.

4 EXAMPLE APPLICATION AND RESULT EVALUATION 4.1 Problem description An algorithm implementation and numerical analysis using the proposed model and solution method for a 10-year bi-objective pavement M&R scheduling problem are carried out. The data used in this example problem is collected from the Dallas District’s Pavement Management Information System (PMIS). In this PMIS, there are five different functional classes of roads: Business Road (BR), Farm to the Market (FM), Interstate Highway (IH), State Highway (SH), and US Highway (US). Because of their similarities in terms of deterioration patterns, those roads are allocated into three broader groups as shown in Table 2. In the database, the length of each section is around 1.6 km (1 mile).

282


Table 2 Pavement length Road group

Table 4 M&R treatment effects

Length (lane-kilometers)

Group I (IH, US, and BR) Group II (SH) Group III (FM)

M&R treatment category

8,299 3,104 5,045

Do Nothing

Table 3 Pavement initial condition (percent) Preventive Maintenance

Road group Condition state Very Good (100–80) Good (80–60) Fair (60–40) Poor (40–20) Very Poor (20–0)

Group I

Group II

Group III

73 11 7 5 4

58 15 10 9 8

62 16 10 8 4

Rehabilitation

Reconstruction

The PMIS stores three scores that represent the general condition of a road pavement (TxDOT 2000): distress score (DS), ride score (RS), and condition score (CS). DS reflects the amount of visible surface deterioration of a pavement. It ranges from 1 (the most distress) to 100 (the least distress). RS is a measure of the pavement’s roughness. It ranges from 0.1 (the roughest) to 5.0 (the smoothest). CS represents the pavement’s overall condition in terms of both distress and ride quality (serviceability index values). It ranges from 1 (the worst condition) to 100 (the best condition). In this case study, the CS is adopted as the pavement performance indicator. Furthermore, the condition of road pavement is defined as five condition states in terms of their CS values: Very Good (100–80), Good (80–60), Fair (60–40), Poor (40–20), and Very Poor (20–0). The initial conditions of the road network are shown in Table 3. The numbers in Table 3 represent the percentage of a certain road group in a specific condition state. For example, 73% of the group I road pavements are in the “Very Good” condition states. As can be seen in the table, the majority of the road network is in the “Very Good” condition state. In Texas, the current statewide pavement condition goal, set by the Texas Transportation Commission in 2002, is a single-tier, one-size-fits-all objective: a lanemile of high-traffic, metro, interstate highway has the same impact on the condition score as a lane-mile of low-traffic, rural, farm-to-market road. In this case study, the goal of the M&R scheduling analysis is to maintain 90% of the road pavements in the “Very Good” condition in the next 10 years. M&R treatments can be selected from multiple levels, from the simplest and cheapest one in the

Condition state before treatment

Condition state after treatment

Very Good Good Fair Poor Very Poor Very Good Good Fair Poor Very Poor Very Good Good Fair Poor Very Poor Very Good Good Fair Poor Very Poor

Very Good Good Fair Poor Very Poor Very Good Very Good Fair Poor Very Poor Very Good Very Good Very Good Good Fair Very Good Very Good Very Good Very Good Very Good

preventive maintenance to the most complex and expensive one in the rehabilitation. Preventive maintenance, including fog seal, slurry seal, and rejuvenation, is aimed to extend the life of bituminous surfaces by retarding the effects of weathering and aging before significant amounts of distress have occurred. Rehabilitation represents resealing and overlays with different thicknesses. M&R treatments are assumed to be applied at the beginning of each year. However, it is not necessary for programming at the network level to be as detailed as at the project level. Therefore, four simplified M&R treatment levels are assumed in this case study (Table 4). For every section at any given year, one of the four possible treatments should be performed. Associated with each M&R treatment is a set of effectiveness and costs. In this case study, the effectiveness values are chosen by considering the work of Smilowitz and Madanat (2000). The effectiveness of M&R treatments for a pavement section is listed in Table 4. The cost of all types of treatments (see Table 5) is selected by considering the research conducted by Wang et al. (2003). The determination of the values in Tables 4 and 5 is for the purpose of demonstrating the proposed methodology. When applied to practice, the pavement agencies should choose the appropriate M&R effectiveness and cost functions according to their experience.


Table 7 Sample iterations generated by the parametric method

Table 5 M&R treatment costs

M&R treatment

M&R treatment unit cost ($1,000/km)

Do Nothing Preventive Maintenance Rehabilitation Reconstruction Do Nothing Preventive Maintenance Rehabilitation Reconstruction Do Nothing Preventive Maintenance Rehabilitation Reconstruction

0 10 100 500 0 8 80 400 0 5 20 100

Road group I

II

III

283

Parameter set Iteration 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

The deterioration transition probabilities of each road group are calculated based on the historical data from the Dallas District’s PMIS and is presented in Table 6. The elements in Table 6 represent the annual deterioration rate of pavement sections. For example, for a section in Group I, if the current condition state is “Very Good,” with a 10% probability it will deteriorate to the state of “Good” within 1 year.

Objective pair

w1

w2

z1

z2

Combinedobjective value

0.000 1.000 0.997 0.992 0.998 0.980 0.996 0.997 0.999 0.976 0.984 0.996 0.997 0.997 0.998 0.998 0.999 0.973 0.979 0.982 0.988

1.000 0.000 0.003 0.008 0.002 0.020 0.004 0.003 0.001 0.024 0.016 0.004 0.003 0.003 0.002 0.002 0.001 0.027 0.021 0.018 0.012

−0.294 −0.914 −0.645 −0.522 −0.801 −0.429 −0.583 −0.738 −0.867 −0.369 −0.493 −0.558 −0.603 −0.692 −0.774 −0.839 −0.900 −0.338 −0.399 −0.465 −0.513

0.000 191.107 45.238 11.232 105.949 5.471 26.879 77.439 146.340 2.701 8.861 19.926 32.634 60.602 92.617 128.022 173.809 1.519 4.058 7.298 10.336

0.000 −0.894 −0.496 −0.432 −0.605 −0.311 −0.485 −0.537 −0.673 −0.294 −0.343 −0.479 −0.492 −0.515 −0.568 −0.630 −0.717 −0.288 −0.304 −0.322 −0.379

Figure 3 shows the frontier curve formed by the obtained nondominated Pareto-optimal solutions of problem P3 by setting budget constraint Bt = $200 million [Equation (15)] and the condition requirement constraint X∗ = 0% [Equations (16) and (17)]. Figure 4 presents a similar graph by setting Bt = $150 million and X∗ = 60%. The negative sign before z1 is generated by converting a maximization problem (P1) into a minimization problem (P3). As shown in Figures 3 and 4, any gain in one objective is involved with a loss in the other one, and the rates of gain or loss in objectives change along the frontier curve. It is left to the

4.2 Result analysis The parametric method is coded and tested in MATLAB R14 and all our numerical experiments are performed on a desktop computer with a 3.4 GHz CPU and 1 GB of memory. Table 7 presents a list of 20 initial Pareto-optimal solutions, as examples, generated by the parametric method, including the first two solutions that correspond to a pair of prespecified initial parameter sets and their subsequent solutions generated by the iterative procedure.

Table 6 Transition probability matrix of road groups Next state Group I

Group II

Group III

Initial state

Very Good

Good

Fair

Poor

Very Poor

Very Good

Good

Fair

Poor

Very Poor

Very Good

Good

Fair

Poor

Very Poor

Very Good Good Fair Poor Very Poor

0.85 0.00 0.00 0.00 0.00

0.10 0.57 0.00 0.00 0.00

0.03 0.28 0.47 0.00 0.00

0.01 0.12 0.39 0.56 0.00

0.00 0.04 0.13 0.44 1.00

0.74 0.00 0.00 0.00 0.00

0.16 0.35 0.00 0.00 0.00

0.07 0.37 0.45 0.00 0.00

0.03 0.21 0.44 0.55 0.00

0.01 0.07 0.11 0.45 1.00

0.77 0.00 0.00 0.00 0.00

0.14 0.36 0.00 0.00 0.00

0.06 0.39 0.38 0.00 0.00

0.03 0.19 0.43 0.41 0.00

0.01 0.06 0.19 0.59 1.00

284


200

180

(z2) Average Annual M&R Cost (million $)

160

140

120

100

80

60

40

20

0

0

10

20

30

40

50

60

70

80

90

100

(-z1) Average Proportion of Road Network in the "Very Good" Condition State (%)

Fig. 3. The frontier curve of nondominated solutions for Bt = $200 million and X∗ = 0%.

200

180

(z2) Average Annual M&R Cost (million $)

160

140

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(-z1) Average Proportion of Road Network in the "Very Good" Condition State (%)

Fig. 4. The frontier curve of nondominated solutions for Bt = $150 million and X∗ = 60%.


Table 8 Comparison between parametric method and weighting method Computational time (seconds)

Number of identified nondominated solutions

Number of Parametric Weighting Parametric evaluations method method method 100 200 500 1000

10.23 20.39 63.19 140.22

9.21 19.84 50.32 120.73

94 180 415 897

Weighting method 63 132 230 549

decision maker’s interest and responsibility to choose one pair of (−z1 , z2 ) from the set of available alternative solutions, subject to other project restrictions and societal constraints. Compared with Figure 3, the curve in Figure 4 lies in a much narrower range. This is due to the lower budget constraint and higher condition constraint. By changing the values of Bt and X∗ , different frontier curves can be obtained. 4.3 Performance comparison A comparison is also carried out between the parametric method and the weighting method. The latter appears to be the most frequently used solution method for bi-objective optimization problems in the literature. Different from the parametric method, the basic idea of the weighting method is simply to first specify scalar weight factors for each objective to be optimized, and then adds them together into a single function that can be solved by any single-objective optimizer. The weight factors of the weighting method are determined as a priori and each of the factors is set with its value at equal intervals between 0 and 1. The major difference between the parametric method and the weighting method is that weight factors are calculated at each iteration in the parametric method while the factors are set as a priori in the weighting method. To make a fair comparison, the number of evaluations of both the parametric method and the weighting method is set to the same. The comparison result is given in Table 8. The computational performance of these two methods is compared at four different numbers of evaluations. The weighting method runs slightly faster than the parametric method since the former method does not have an extra parametric calculation procedure like the latter. However, the parametric method generates many more nondominated solutions. If the computing time per generated

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nondominated solution is used as the efficiency indicator, it is clear that the parametric method outperforms the weighting method. Moreover, the difference of computing time per solution between those two methods seems to be quite small, given that we only used three road groups of one city in this case study. However, the time difference would be much bigger when applied to state-level scheduling problems where more road groups are presented.

5 CONCLUSIONS AND FURTHER APPLICATIONS Preserving network-level road pavements typically involves decisions about how, where, and when to maintain and rehabilitate to keep the pavement conditions at a reasonable level using the limited budget. In this article, a bi-objective LP problem is presented for network-level pavement M&R scheduling. The parametric method is applied to solve this problem for the complete Pareto-optimal solution set. The solution method is numerically and comparatively evaluated through a real-world case study. The evaluation result shows that the parametric method is a more attractive solution approach for the bi-objective pavement M&R scheduling problem than existing methods, such as the weighting method: (1) it can generate the complete set of Pareto-optimal solutions; (2) it is much more efficient in identifying Pareto-optimal solutions in terms of the computing time per solution. These advantages are particularly important to large-scale problems. With such an exact and more efficient solution method, the decision maker is able to more quickly make trade-off decisions while avoiding the possibility of ignoring any potential attractive Pareto-optimal solutions. Finding the Pareto-optimal frontier is just the first step in the complete pavement M&R scheduling decision-making process, however. The decision maker needs to subsequently pick up a best compromise solution from the Pareto-optimal frontier between cost and performance subject to other project restrictions and social constraints. Given the selected best solution, the proportion of each road group that needs a certain M&R treatment can be accordingly determined. Finally, the selection of specific sections to receive the treatment can be concluded in terms of other considerations, for example, traffic volume. The parametric method is potentially applicable to evaluating multiobjective M&R scheduling problems or multiobjective M&R scheduling problems of integer type. In particular, when the number of objectives is equal to or greater than 3, the presented bi-objective parametric method may not be directly

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