A Compromise Programming Model for Highway Maintenance Resources Allocation Problem

This paper formulates a bilevel compromise programming model for allocating resources between pavement and bridge deck maintenances. The first level of the model aims to solve the resource allocation problems for pavementmanagement and bridge deckmaintenance, without considering resource sharing between them. At the second level, the model uses the results from the first step as an input and generates the final solution to the resource-sharing problem. To solve the model, the paper applies genetic algorithms to search for the optimal solution. We use a combination of two digits to represent different maintenance types. Results of numerical examples show that the conditions of both pavements and bridge decks are improved significantly by applying compromise programming, rather than conventional methods. Resources are also utilized more efficiently when the proposed method is applied.


Introduction
By the end of the 1960s, major construction of highway networks ended in developed countries such as the US and Canada. The main tasks of highway agencies shifted from planning, design, and construction to management and maintenance of the highway infrastructure, which usually consumes large resources. In the US, for instance, the total investment in highway infrastructure has reached $1 trillion 1 . In recent years, highway agencies have to face tough budgeting problems in maintaining existing highway facilities. Due to ever-increasing transportation demand, more funds are needed to rehabilitate the deteriorating highway facilities, and the gap between required funds and budgets is increasing. It becomes critical that the limited resources should be more effectively allocated and used. This paper applies the concepts of compromise programming and formulates a bilevel model for the resource allocation problem. The first level of the model solves the resource allocation problems for pavement management and bridge deck maintenance, without considering resource sharing between them. The second level of the model solves the resources allocation problems considering resource sharing. The following content is organized as below. Section 2 presents the bilevel optimization model based oncompromising programming. Section 3 is a numerical example applying genetic algorithms to solve the model. The conclusions and recommendations for further study are presented in Section 4.

Model
The basic concepts of CP are presented below. Consider the following general multiobjective problem 13, 14 : The constraints can be expressed in a general form as x ∈ X {x ∈ R n | g j x ≤ 0, j 1, 2, . . . , q; h l x 0, l 1, 2, . . . , r}, where q and r are numbers of inequality and equality constraints, respectively.
If the problem does not have any conflicting objectives, one can apply conventional optimization approaches to obtain a final solution that optimizes all objective functions. However, the objectives are mutually conflicting in many real-world engineering problems. In addition, those objectives can have different measurement scales. Zeleny 4 developed a method that transforms these objectives into a set of comparable scales. He then formulated a new single-objective optimization problem as in 2.2 and showed that the optimal solution from the new problem is a Pareto-optimal solution to the original problem 2.1 . Consider x ∈ X, λ ∈ Λ.

2.2
In the Previous expression, y 0 i is the optimal solution to the original problem containing only the ith objective function f i x . p is a parameter satisfying 1 ≤ p < ∞.

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The new objective function, L p , is a weighted average of all shortest distances of objective values. L p is called the Manhattan distance.

2.4
In this case, the new objective function, L p , is a weighted summation of geometric distances. When p 2, L p is a Euclidean weighted distance.
c If p ∞, then 2.2 can be transformed into 2.5 . In this case, the new objective function, L p , is a Chebyshev distance as follows x ∈ X, λ ∈ Λ.

2.5
Under such transformations, the multiobjective problem in 2.1 has an equivalent single-objective problem, which can be expressed as x ∈ X, λ ∈ Λ.

2.6
Earlier approaches to highway infrastructure management were usually developed for single facility management. In this section, we present an optimization model for resource allocation in pavement maintenance. A model for bridge deck maintenance is very similar. The major difference between them is that in the pavement maintenance model, road surface is divided into segments of different lengths, while in the bridge deck maintenance model a bridge deck is divided into different areas. In addition, performance indices are different between the two models.
The objective for the pavement maintenance is to maximize the weighted average surface performance after maintenance. It can be formulated as where PQI ia is the pavement quality index of segment i after a type a maintenance, a 0, 1, . . . , A, L i is the length of segment i, ω i is the associated weight for segment i, I is the total number of road segments in the whole network, A is the total number of maintenance types, X 0 ia is a binary decision variable. X 0 ia 1 when a type a maintenance is selected for segment i, and 0 otherwise.

5
Four sets of constraints are considered in the problem: funding constraints, human resource constraints, equipment constraints, and constraints on road maintenance types.
Funding constraints are expressed in 2.8 . It indicates that the total maintenance expenses for all segments must not exceed the total available funds in a specific planning year as follows: where C p ia in the above expression is the maintenance expense when a type a maintenance is selected for segment i and B p is the total available funds for all pavement of the network.
Similar to the funding constraints, human resource constraints can be expressed as where m p iau is the required amount of type u labor when a type a maintenance is selected for segment i and M p u is the total available amount of type u labor. The equipment constraints can be expressed as where e p iak is the required amount of type k equipment when a type a maintenance is selected for segment i and E p k is the total available amount of type k equipment. It is also assumed that in a maintenance cycle, only one type of maintenance is selected for one segment. Thus, one has the following constraints on road maintenance types: Typically, highway assets include pavement and bridges. The objective of highway asset maintenance management is to maximum the overall quality of the entire highway network. In this paper, we only consider two types of maintenance: pavement and bridge deck maintenance. Based on the concepts of compromise programming, we formulate a single-objective function as a weighted summation of two distances as below: where parameters λ 1 , λ 2 represent the weights of pavement and bridge decks, respectively, and ω i , ω j are the weights of pavement segment i and bridge deck unit j, respectively; X ia and Y jr are binary decision variables. X ia 1 if a type a maintenance is selected for pavement segment i, and 0 otherwise; Y jr 1 if a type r maintenance is selected for bridge deck unit j, and 0 otherwise. In 2.12 , we take p as 2 to avoid cancellation of positive and negative values in the objective function. The problem has five sets of constraints, which are discussed in the following content.

Funding Constraints
As the funds for pavement maintenance and bridge deck maintenance can be shared, the total expenses for both pavement and bridge deck maintenance must not exceed the total available funds. Thus, one has the following: where B is the total available funds for entire network maintenance.

Human Resource Constraints
In highway maintenance, labor can be classified into three groups: personnel only capable of managing pavement, personnel only capable of managing bridge decks, and personnel having both capabilities. Among these three groups, personnel in the third group can be shared between pavement maintenance and bridge deck maintenance. Equations 2.14 , 2.15 and 2.16 describe constraints on labor in these three groups, respectively: where f is the total number of labor types that can be shared between pavement maintenance and bridge deck maintenance; U is the total number of labor types that can only be used for pavement maintenance; V is the total number of labor types that can only be used for bridge deck maintenance.
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Equipment Constraints
Equipment resources can also be classified into three groups: equipments only usable in pavement maintenance, equipments only usable in bridge deck maintenance, and equipments usable in both pavement and bridge deck maintenance. Equations 2.17 , 2.18 , and 2.19 describe constraints on equipment in these three groups, respectively: where g is the total number of equipment types that can be shared, K is the total number of equipment types that can be used in pavement maintenance, and L is the total number of equipment types that can be used in bridge deck maintenance.

Constraints on Maintenance Types
In each maintenance cycle, it is assumed that only one type of maintenance can be selected for a pavement segment or bridge deck unit. Hence, we have additional constraints as follows:

Least Performance Constraints
From the transformations of objective functions in compromise programming, it is noted that when parameter p is an even number, the value within the parentheses in 2.12 can be either positive or negative. When this value becomes negative, highway asset maintenance management will not achieve any benefits. Hence, we have additional constraints that the performance of highway infrastructure under resource sharing must exceed or at least equal that without resource sharing. Such constraints can be presented in the following: Y 0 jr S j ω j CR jr ≥ 0.

Numerical Analysis
In this numerical example, the highway network consists of four types of facilities: 16.7 lanekm of state highways, 30.7 lane-km of provincial highways, 59.6 lane-km of county roads, and 20.3 lane-km of bridges. For each pavement unit or bridge deck unit, one can choose from the following four options: major maintenance, moderate maintenance, minor maintenance, and no maintenance. The resources e.g., funds, human resources, equipments amounts required for different maintenance options and for different facility types are assumed to be known. It is also assumed that the information on infrastructure quality after different types of maintenance is available for analysis. Table 1 lists the amounts of human resources required in different types of maintenance. Table 2 shows the productivities for different maintenance types against facility types. In addition, the model requires other information such as unit maintenance expense, equipment allocation, before-and-after facility quality, interest rates, total available funds, and other resources, least performance requirements for each facility unit, and variations of traffic volumes. The input information for each bridge deck unit is similar to that for a pavement segment.
Although the highway network is not large, the space of all feasible solutions to the problem is very large, and complete enumeration is computationally infeasible as a method of finding the optimal solution. In this paper, we apply genetic algorithms GA to search for the optimal solution.
A chromosome in GA is a string of 0-1 numbers. Here we use a combination of two digits to represent various maintenance types. For instance, "00" stands for "no maintenance", while "01," "10," and "11" stand for minor, moderate, and major maintenances, respectively.
In this example, a blue cell represents a pavement segment, and a green cell represents a bridge deck segment. A chromosome, that is, a feasible solution to the model, consists of 10 fractions. Each fraction represents a one-year pavement and bridge deck maintenance plan, see Figure 1. As for any specific fraction, there are 2 parts, blue one and green one. The blue part indicates pavement maintenance plan and the green part corresponds to bridge deck maintenance plan for a given year. · · · · · · · · · · · · · · · · · · · · · · · · · · · 1st year 2nd year 10th year Figure 1: GA code for the solution. There are two steps in solving the problem. At the first step, we obtain the optimized solutions for pavement maintenance and bridge deck maintenance, without considering resource sharing. The second step uses the results from the first step as input and generates the final solution to the resource sharing problem.
The GA parameters are determined with some preliminary analyses. In solving the pavement maintenance problem at the first step, the population size and iteration number, mutation rate are set at 500, 600, and 0.02, respectively. In solving the bridge deck maintenance problem, the population size, iteration, mutation rate to 300, 200, and 0.02, respectively. At the second step computation with resource sharing , the values of these 3 parameters are set at 500, 2000, and 0.03, respectively. Table 3 shows the human resources utilization in the base year for maintenances with and without resource sharing. It is apparent that the utilization efficiency for the six types of labors is increased, respectively. Table 4 compares equipments utilization in maintenances with and without resource sharing. Figure 2 indicates the performance changes of pavement and bridge decks after maintenance. It can be easily seen that for all planning years, both pavement performance and bridge deck performance are improved with resources sharing.

Conclusions
The objective of highway asset maintenance management is to maximize the overall conditions of the highway system after maintenance with limited resources. This is achieved through integrated management for different types of highway facilities. The paper presents a new approach for optimal allocation of highway maintenance resources based on compromise programming. With the concepts of compromise programming, one can consider the relative importance of different highway facilities by introducing associated weights into the objective function. A bilevel model has been developed to analyze the  resource allocation problem in pavement and bridge deck maintenance. Two scenarios in maintenance activities are analyzed: first without and then with sharing of resources between pavement maintenance and bridge deck maintenance. With a robust GA searching for an optimal solution to the problem, it is found that the performance of pavements and bridge decks improves significantly under resource sharing. The maintenance resources e.g., funds, labor, and equipment are utilized more efficiently in the resource-sharing scenario. The results of experimental analyses clearly show the promising features of the model in solving complex resource allocation problems in highway maintenance management. The method developed in this paper is useful to highway agencies in their decision-making activities such as developing maintenance programs, budgeting, and resource allocation.