- Email: [email protected]
Capital budgeting problems in pavement maintenance
CAPITAL BUDGETING PROBLEMS IN PAVEMENT MAINTENANCE?_ RONALD Center
for Cybernetic
Studies,
D.
ARMSTRONG
The University
of Texas. Austin,
TX 78712, U.S.A
and WADE Faculty
of Administrative
Studies.
D.
COOK
York University,
Toronto,
Ontario
M3.l 2R6, Canada
Abstract--This paper models a class of capital budgeting problems in pavement maintenance as nonlinear discontinuous programming problems and presents a solution procedure. The objective of the models developed is to determine if certain road segments should be rehabilitated in the given year (resurfaced, reconstructed or patched), or major maintenance postponed for at least another year. If the road is to be rehabilitated, some variation in funds expended is possible. The marginal return within the allowable interval of variation is estimated to be nonlinear. Two basic models are presented. The first assumes the rehabilitative alternative or design is fixed, and hence the problem has a single linear constraint imposed by the budget. The second is the design selection problem and contains multiple choice constraints in addition to the budget constraint. An illustrative example is included.
1. INTRODUCTION
budgeting problems in the operations research literature are frequently formulated as zero-one linear programming problems [S, 61. However, it may be the case that the ‘go/no go’ restrictions dictated by binary variables are too abrupt to be practical. Often a project is either not funded or is funded at levels of expenditure over some interval. The problem discussed herein is of such a nature. Section 2 of this paper presents a detailed description of a problem involving the allocation of funds toward the maintenance of paved roads in a given area of the country. The model developed in section 3 is not the usual binary type since a road is either repaved in a given year with a certain amount of flexibility in the funds expended, or else major maintenance is deferred for at least another year. This built-in variation in the amount of funds allocated to a project leads to a ‘quasi-integer’ programming formulation. Armstrong and Sinha [2] have given a branch and bound algorithm to handle such quasi-integer models when returns within the allowable capital expenditure interval are linear. Such results are not directly applicable to the pavement maintenance problem since additional expenditure implies a higher quality road, and the marginal value of expenditures beyond a base level is estimated to be nonlinear. Hence, the model is a quasi-integer programming problem with nonlinear returns and a single linear budget constraint. Section 4 presents a solution procedure for the models of section 3, together with an illustrative example. CAPITAL
2. A
PAVEMENT
MANAGEMENT
SYSTEM
In both the U.S.A. and Canada, highway authorities are responsible for initiating effective maintenance strategies on pavements when required. Pavement maintenance is an area where millions of dollars are spent annually on the correction of defects. In addition, vast amounts of money, time and manpower have been expended in conducting controlled experiments with both rigid and flexible pavements in an attempt to quantify their performance over time and under various rehabilitative alternatives, axle loadings, environmental conditions,. . . etc. Studies conducted during the past 15 yr t This research
was partly
supported
by NSF Grant 175
MSC77-00100
and NRC
Grant
A8966.
176
RONALD D. ARMSTRONGand WADE D. COOK
in the U.S.A., Australia, Kenya and Brazil (see for example [4], [S] and [IO]) have provided information sources which are the principal inputs to decisions relating to highway rehabilitation. One highway management system in particular with which the authors are familiar is the Ministry of Transportation and Communications, Province of Ontario, Canada. This organization maintains a vast inventory (data bank) on the condition of all roads in the province. The available data relates to both structural and geometric adequacy of pavements. Structural adequacy is a function of cracking, roughness, frost heaving, settling, flushing and numerous other distortions. Geometries relate to lane width, slope. crossfall standards, shoulder width,. . . etc. Pavements have in the past been rated via the use of a variety of indices, with the most relevant one being the so-called pavement condition rating (PCR). This rating is to some extent all-encompassing, but for all practical purposes is a measure of structural adequacy. (It should be noted that structural adequacy is also a function of such things as crossfall standards which was earlier categorized under the heading of geometries). In an attempt to develop models for optimal pavement management, data was collected from historical records on different categories of roads. The major categories of interest are: (1) traffic level (levels examined were ~2000 vehicles per day, 200&4000, 400&6000, . . . etc); (2) region of the province (Northwest, North, South, East and Central regions); and (3) rehabilitative strategy used (1 lift of hot mix asphalt, 2 lifts, 3 lifts, 4 lifts, reconstruction, recycling and hot mix patching). A plot of data (PCR versus age of pavement since last major maintenance) in each category revealed a nonlinear trend. A sample pavement performance curve is given in Fig. 1. A family of functional forms which appear to fit the data best, and which were used in the analysis is the set of all exponentials of the form P, - P = KAfi. PO is the rating immediately after rehabilitation (say, a 2 lift resurfacing), P is the rating A years after resurfacing, and K and fi are regression coefficients. This family of curves was used in analyzing pavement performance in the AASHO road tests [IO]. For the problem at hand a (K, /I) pair was determined for each category of road. In some instances once the highway has been placed into a given category according to traffic number of lanes and region, the design (that is, the particular rehabilitative alternative of the ones listed ,above) is fixed. In this case the aspect of rehabilitation which isn’t predetermined has to do with variation within a design. That is. once a design has been decided upon, up to 10 or 15% over the minimum cost necessary to meet standards can be spent on improving the future performance of a highway. Alternatively, there may be a choice of rehabilitative strategies as well as within-strategy PCR
I
2
I
1
4
Fig.
6
I
Age (years)
Capital
variations and other
budgeting
to consider. The models considerations pertaining
177
in pavement maintenance
problems
in the following section attempt to pavement management.
3. MODELS FOR CAPITAL BUDGETING PAVEMENT MAINTENANCE
to deal with these
IN
Numerous criteria exist for evaluating pavement performance; user cost, road roughness, average PCR (per year) and total PCR over the life of the pavement are examples. For purposes of this paper we use the area under the PCR curve as a measure of worth of a pavement for a fixed analysis period of T years (e.g. T = 10) with some allowance for salvage to be made after that time. A jixed
design model
Assume n projects are under consideration for some form of major rehabilitation in the current year. These projects may span the spectrum of traffic categories, various regions and have diverse historical records of performance. Let the decision variable xj denote the level of expenditure to be assigned to project j where j E -(1.2,. . . . n ). For different levels of expenditure xi on project j different performance curves arise; that is, for each xj, appropriate K and fi values can be determined. We assume xj can lie in the range [l. uj] (xj has been scaled by the minimum positive allocation). The worth (benefit) of the pavement is then given by fj(Xj) = s,’ [P(Xj) + K(Xj)A”(XJ)] dA + Sj(Xj)
(3.1)
where Sj(xj) is the salvage value. There are many different ways of measuring the salvage value of a road. Generally, Sj(xj) should represent the long run worth of the road following the analysis period which ends at time T. Perhaps the most realistic approach in measuring Sj(Xj) is to assume that an optimal maintenance procedure will be used after T. Then using standard discounting methods over a 20 or 30 yr period following T, take Sj(xj) to be the best value which can be attained subject to budget restrictions. If xi = 0 project .j is not programmed; otherwise, 1 < xi d uj. We therefore, wish to solve the quasi-integer programming problem: Maximize
f
fj(xj),
j:l
subject
to (3.2) 1 < Xj < Uj or
Xj
=
0, j = 1,2,. . , n,
where B is the total current year budget and aj is the minimum nonzero expenditure on project .j. The concave [7] functions fi(xj) take a form similar to that shown in Fig. 2. A quadratic provides an appropriate fit to the points generated from (equation 3.1), so that the form offi assumed in (equation 3.2) is hjx; fjCxj)
=
o
+ cjxj + dj
for Xj E [ 1, uj] otherwise
In a later section we provide an example illustrating the use of these Section 4 discusses a solution procedure for (equation 3.2).
functions.
A variable design model
In the previous model to be known in advance.
the design or alternative for a given project was assumed For project j,, for example, the selected alternative might
178
RONALD D. ARMSTRONG and WADE D. COOK
-I---
Fig. 2
was, therefore, that be 2 lifts of hot mix asphalt; for ,jz, 1 lift;. . etc. The assumption a ‘best’ design could be selected in advance. Let us now bring the design selection problem directly into the model. Denote the allowable designs for project j by I(j). 1(j) will in general be a proper subset of the set of all possible designs since, for example, 1 lift of asphalt would never be applied to high traffic roads. Similarly, hot mix patching would seldom be a strategy for a road at a very low PCR. Let xij represent the level of expenditure on project j if design i is applied. Accordingly, let Jj(_uij) represent the quadratic benefit function for project j and design i. The design selection problem is: Maximize subject
i
I:
j=l
i-l
fij(xij)
to (3.3) xijxkj=O;i=l
,....
1(j)-l;k=i+l,...,
r(,j);,j=l,...,
11
1 6 xij Q uij or xii = 0 for all i, j. If Xij # 0 then design i is used on project ,j. It is noted that (equation 3.3) is a generalization of (equation 3.2) in that I(j) contains exactly one design in the latter case. In the following section we discuss an algorithm for solving problems of the form presented above. 4. ALGORITHMIC
PROCEDURES
The most direct approach for solving the quasi-integer knapsack (equation 3.2) is to discretize the problem over the feasible region. Let Pki. ,j = 0. 1,. , K(j); ,j = 1, 2,. . . , n, represent given points satisfying XoJ = 0 and 1 < Xkj 6 uj for k # 0. Problem (equation 3.2) can be approximated by the following multiple choice knapsack problem.
j=l
subject
k-l
to
k&;.kj
= I,
j = 1,2 ,...,
ikj = 0 or 1.
n
Capital
budgeting
problems
in pavement
maintenance
179
Sinha and Zoltners [9] present a solution algorithm for solving this problem. Through repeated refinements of the approximation process, or by choosing a very small grid originally, an e-optimal solution for problem (3.2) can be obtained. Armstrong et al. [l] present a branch-and-bound algorithm for solving (equation 3.2) directly. A convex relaxation of the problem is solved at each node and the dichotomy performed restricts xj to be 0 down one branch and to be in [l, Uj] down the other branch. List structures similar to those used successfully by Barr and Ross [3] to solve the zero-one knapsack problem are present in the computer code implementation of the algorithm. This code is available, for academic use, from the authors upon request. An extension of the algorithm from [l] to solve problem (3.3) is currently under study by the authors. However, as yet, no computer code is available. The discretization procedure previously described for problem (3.2) can also be utilized on problem (3.3) to create a multiple choice knapsack problem. Here, we let Xkij, represent
the selected
K(i, j); ,j = 1,2,. . . , n; i = 1, 2, . . . I(j)
k=O,l,..., approximation -Uoij = 0
The problem
points. 1 6 xkij < uij, 1 < k < K(i,,j)
and
becomes: I(j)
Maximize
i j=l
subject
1 i=l
Wi.3
x
f;j(~,j)&.ij
k=l
to
I(j)
1 i=l
Wi,j)
1
E.kij = 1;
j = 1,2,.
. . , n,
k=O
~kij = 0 or 1
5. SAMPLE
PROBLEM
A sample of 400 projects was selected from records of the Ontario Ministry of Transportation. Associated with each project was a capital outlay x and, utilizing the subsequent performance related to each, f(x) was computed. This data was split into 20 categories according to the average daily traffic (ADT) ((2000 ADT, 200&4000 ADT. 4000-8000 ADT and > 8000 ADT) and region of the province (Northern, Northwestern, Eastern, Central and Southwestern). The number of data points varied from region to region. A regression was performed on the data in each of the categories, using a quadratic model which seemed to provide a good fit. The results of the regression are shown in Table 1. These values provide the objective function parameters for the quasi-integer knapsack model problem (3.2). In the particular example solved as an illustration, the total budget available was taken as B = $264,590. The minimum expenditures aj are also given in Table 1. In the example we have taken each of the 20 functions as representing a specific project. This 20 project problem was then solved using the algorithm of [l]. The results are shown in Table 2. In the general case with many projects there would be several in each category. All projects in the same category would be represented by the same quadratic, on a per mile basis. However, the scaling parameter. the minimum expenditure and the value .f(x) (which is a function of length of the road section) would depend on the particular project in that category.
1. Data
-5.0
27.1
25.5
31.8
400& 8000
>8000
-2.1
-3.6
b -3.0
2F.4
9.4
8.3
12.0
G.0
North
-2.84
-3.92
-5.91
d -3.04
28.8
24.8
31.9
3:3
is given from a regional planning model single linear constraint
<2000 2OOtS 4000
Traffic
Table
-4.1
- 3.0
-4.2
b -3.1
9.0
7.0
10.3
28c.o
Northwestern
-4.2
-3.49
-4.79
d -3.35
32.2
36.7
46.1
2:0
-3.8
-4.2
-5.3
b - 2.8 ;.6
9.0
12.0
17.0
Eastern
-4.14
-4.46
-6.27
d 3.33
26.1
33.3
40.3
-2.5
-5.3
-3.8
b ZFY.0 -2.1
F.5
8.7
12.4
12.0
Central
in pavement maintenance for the Province of Ontario. Objective functions has coefficient a (in thousand of dollars). The upper bound on the variables
-2.9
-6.12
-4.58
d -2.17
38.4
19.5
36.7
4r.3
are of the form is 1.1
-2.1
-1.3
-3.6
b -3.9
9.3
4.0
11.0
sc4
Southwestern
the
-2.7
-1.37
-3.9
d -4.55
hx* + cx + d and
S
H
n
fr
;
g
P > E 2 g z
Capital Table Traffic < 2000 200&4000 400&8000 > 8000
budgeting
problems
2. This table gives the optimal North 1.1 0 0 1.09
in pavement
solution
Northwestern 1.1 0 0 0
to the model Eastern 0 1.1 0 0
181
maintenance described Central
I .04 0 0 1.1
in Table
2
Southwestern 0 0 1.0 1.0
6. CONCLUSIONS
The problem of maintaining high quality roads with limited funds is encountered continually by highway authorities throughout the world. As additional roads are constructed and traffic frequency increases, this problem is becoming more complex. Certain steps have been taken to quantify aspects of pavement maintenance. Controlled studies [3, 12, 131 have provided reasonable approaches to measuring the value of a road and estimating deterioration. As the result, data and pavement conditioning rating functions are available as input to mathematical programming models which are designed to budget capital to pavement maintenance projects. Two such models are presented in this paper. It is hoped that one contribution of this paper will be to provide a framework for future modeling research in the area of pavement maintenance. Valuable input to the decision making processes of highway authorities can be derived through a utilization of the knapsack models of this paper. Obvious extensions of models presented in this paper are to consider budgeting in future time periods and to enter restrictions on the number of roads resurfaced in a given subarea. The majority of these model extensions can be constructed as, or at least approximated by, integer programming problems. For certain classes of these problems, such as those presented herein, efficient solution procedures exist. For the more complex problems, one should expect to expend a great deal of computer time to obtain an optimal solution or be satisfied with a suboptimal solution. REFERENCES 1. Armstrong R. D., Cook W. D. and Palacios-Gomez F. An Algorithm for a Quasi-integer Nonlinear Knapsuck Problem. Working Paper, The University of Texas at Austin. 2. Armstrong R. D. and Sinha P. An application of quasi-integer programming to menu planning with variable portion size. Mgmt Sci. 21, 474482 (December 1974). 3. Barr R. S. and Ross G.-T. A Linked Lisr Data Structure f;,r a Binury Knapsack Algorithm. Center for Cybernetic Studies, Report 232 (August 1975). 4. Ciaffey P. .I. Running costs of Motor Vehicles as affected by Road Designs and Trujic. The National Academy of Sciences (1971). 5. Garfinkel R. S. and Nemhauser G. L. Integer Programming, Wiley, New York (1972). 6. Geoffrion A. and Marsten R. E. Integer programming algorithms: a framework and state-of-the-art survey. Mgmt Sci. 18, 465-491 (1972). I. Mangasarian 0. L. Nonlinear Programming, McGraw-Hill, New York (1969). 8. Pelensky E. The Cost of Urban Car Traoel. Ontario Ministry of Transportation and Communications (1970). 9. Sinha P. and Zoltners A. The multiple choice knapsack problem. To appear in Operations Research. 10. “The ASSHO Road Test,” Highway Research Board of the National Academy of Sciences and the National Research Council, Report No. 5, Washington, DC (1962).
for Cybernetic
Studies,
D.
ARMSTRONG
The University
of Texas. Austin,
TX 78712, U.S.A
and WADE Faculty
of Administrative
Studies.
D.
COOK
York University,
Toronto,
Ontario
M3.l 2R6, Canada
Abstract--This paper models a class of capital budgeting problems in pavement maintenance as nonlinear discontinuous programming problems and presents a solution procedure. The objective of the models developed is to determine if certain road segments should be rehabilitated in the given year (resurfaced, reconstructed or patched), or major maintenance postponed for at least another year. If the road is to be rehabilitated, some variation in funds expended is possible. The marginal return within the allowable interval of variation is estimated to be nonlinear. Two basic models are presented. The first assumes the rehabilitative alternative or design is fixed, and hence the problem has a single linear constraint imposed by the budget. The second is the design selection problem and contains multiple choice constraints in addition to the budget constraint. An illustrative example is included.
1. INTRODUCTION
budgeting problems in the operations research literature are frequently formulated as zero-one linear programming problems [S, 61. However, it may be the case that the ‘go/no go’ restrictions dictated by binary variables are too abrupt to be practical. Often a project is either not funded or is funded at levels of expenditure over some interval. The problem discussed herein is of such a nature. Section 2 of this paper presents a detailed description of a problem involving the allocation of funds toward the maintenance of paved roads in a given area of the country. The model developed in section 3 is not the usual binary type since a road is either repaved in a given year with a certain amount of flexibility in the funds expended, or else major maintenance is deferred for at least another year. This built-in variation in the amount of funds allocated to a project leads to a ‘quasi-integer’ programming formulation. Armstrong and Sinha [2] have given a branch and bound algorithm to handle such quasi-integer models when returns within the allowable capital expenditure interval are linear. Such results are not directly applicable to the pavement maintenance problem since additional expenditure implies a higher quality road, and the marginal value of expenditures beyond a base level is estimated to be nonlinear. Hence, the model is a quasi-integer programming problem with nonlinear returns and a single linear budget constraint. Section 4 presents a solution procedure for the models of section 3, together with an illustrative example. CAPITAL
2. A
PAVEMENT
MANAGEMENT
SYSTEM
In both the U.S.A. and Canada, highway authorities are responsible for initiating effective maintenance strategies on pavements when required. Pavement maintenance is an area where millions of dollars are spent annually on the correction of defects. In addition, vast amounts of money, time and manpower have been expended in conducting controlled experiments with both rigid and flexible pavements in an attempt to quantify their performance over time and under various rehabilitative alternatives, axle loadings, environmental conditions,. . . etc. Studies conducted during the past 15 yr t This research
was partly
supported
by NSF Grant 175
MSC77-00100
and NRC
Grant
A8966.
176
RONALD D. ARMSTRONGand WADE D. COOK
in the U.S.A., Australia, Kenya and Brazil (see for example [4], [S] and [IO]) have provided information sources which are the principal inputs to decisions relating to highway rehabilitation. One highway management system in particular with which the authors are familiar is the Ministry of Transportation and Communications, Province of Ontario, Canada. This organization maintains a vast inventory (data bank) on the condition of all roads in the province. The available data relates to both structural and geometric adequacy of pavements. Structural adequacy is a function of cracking, roughness, frost heaving, settling, flushing and numerous other distortions. Geometries relate to lane width, slope. crossfall standards, shoulder width,. . . etc. Pavements have in the past been rated via the use of a variety of indices, with the most relevant one being the so-called pavement condition rating (PCR). This rating is to some extent all-encompassing, but for all practical purposes is a measure of structural adequacy. (It should be noted that structural adequacy is also a function of such things as crossfall standards which was earlier categorized under the heading of geometries). In an attempt to develop models for optimal pavement management, data was collected from historical records on different categories of roads. The major categories of interest are: (1) traffic level (levels examined were ~2000 vehicles per day, 200&4000, 400&6000, . . . etc); (2) region of the province (Northwest, North, South, East and Central regions); and (3) rehabilitative strategy used (1 lift of hot mix asphalt, 2 lifts, 3 lifts, 4 lifts, reconstruction, recycling and hot mix patching). A plot of data (PCR versus age of pavement since last major maintenance) in each category revealed a nonlinear trend. A sample pavement performance curve is given in Fig. 1. A family of functional forms which appear to fit the data best, and which were used in the analysis is the set of all exponentials of the form P, - P = KAfi. PO is the rating immediately after rehabilitation (say, a 2 lift resurfacing), P is the rating A years after resurfacing, and K and fi are regression coefficients. This family of curves was used in analyzing pavement performance in the AASHO road tests [IO]. For the problem at hand a (K, /I) pair was determined for each category of road. In some instances once the highway has been placed into a given category according to traffic number of lanes and region, the design (that is, the particular rehabilitative alternative of the ones listed ,above) is fixed. In this case the aspect of rehabilitation which isn’t predetermined has to do with variation within a design. That is. once a design has been decided upon, up to 10 or 15% over the minimum cost necessary to meet standards can be spent on improving the future performance of a highway. Alternatively, there may be a choice of rehabilitative strategies as well as within-strategy PCR
I
2
I
1
4
Fig.
6
I
Age (years)
Capital
variations and other
budgeting
to consider. The models considerations pertaining
177
in pavement maintenance
problems
in the following section attempt to pavement management.
3. MODELS FOR CAPITAL BUDGETING PAVEMENT MAINTENANCE
to deal with these
IN
Numerous criteria exist for evaluating pavement performance; user cost, road roughness, average PCR (per year) and total PCR over the life of the pavement are examples. For purposes of this paper we use the area under the PCR curve as a measure of worth of a pavement for a fixed analysis period of T years (e.g. T = 10) with some allowance for salvage to be made after that time. A jixed
design model
Assume n projects are under consideration for some form of major rehabilitation in the current year. These projects may span the spectrum of traffic categories, various regions and have diverse historical records of performance. Let the decision variable xj denote the level of expenditure to be assigned to project j where j E -(1.2,. . . . n ). For different levels of expenditure xi on project j different performance curves arise; that is, for each xj, appropriate K and fi values can be determined. We assume xj can lie in the range [l. uj] (xj has been scaled by the minimum positive allocation). The worth (benefit) of the pavement is then given by fj(Xj) = s,’ [P(Xj) + K(Xj)A”(XJ)] dA + Sj(Xj)
(3.1)
where Sj(xj) is the salvage value. There are many different ways of measuring the salvage value of a road. Generally, Sj(xj) should represent the long run worth of the road following the analysis period which ends at time T. Perhaps the most realistic approach in measuring Sj(Xj) is to assume that an optimal maintenance procedure will be used after T. Then using standard discounting methods over a 20 or 30 yr period following T, take Sj(xj) to be the best value which can be attained subject to budget restrictions. If xi = 0 project .j is not programmed; otherwise, 1 < xi d uj. We therefore, wish to solve the quasi-integer programming problem: Maximize
f
fj(xj),
j:l
subject
to (3.2) 1 < Xj < Uj or
Xj
=
0, j = 1,2,. . , n,
where B is the total current year budget and aj is the minimum nonzero expenditure on project .j. The concave [7] functions fi(xj) take a form similar to that shown in Fig. 2. A quadratic provides an appropriate fit to the points generated from (equation 3.1), so that the form offi assumed in (equation 3.2) is hjx; fjCxj)
=
o
+ cjxj + dj
for Xj E [ 1, uj] otherwise
In a later section we provide an example illustrating the use of these Section 4 discusses a solution procedure for (equation 3.2).
functions.
A variable design model
In the previous model to be known in advance.
the design or alternative for a given project was assumed For project j,, for example, the selected alternative might
178
RONALD D. ARMSTRONG and WADE D. COOK
-I---
Fig. 2
was, therefore, that be 2 lifts of hot mix asphalt; for ,jz, 1 lift;. . etc. The assumption a ‘best’ design could be selected in advance. Let us now bring the design selection problem directly into the model. Denote the allowable designs for project j by I(j). 1(j) will in general be a proper subset of the set of all possible designs since, for example, 1 lift of asphalt would never be applied to high traffic roads. Similarly, hot mix patching would seldom be a strategy for a road at a very low PCR. Let xij represent the level of expenditure on project j if design i is applied. Accordingly, let Jj(_uij) represent the quadratic benefit function for project j and design i. The design selection problem is: Maximize subject
i
I:
j=l
i-l
fij(xij)
to (3.3) xijxkj=O;i=l
,....
1(j)-l;k=i+l,...,
r(,j);,j=l,...,
11
1 6 xij Q uij or xii = 0 for all i, j. If Xij # 0 then design i is used on project ,j. It is noted that (equation 3.3) is a generalization of (equation 3.2) in that I(j) contains exactly one design in the latter case. In the following section we discuss an algorithm for solving problems of the form presented above. 4. ALGORITHMIC
PROCEDURES
The most direct approach for solving the quasi-integer knapsack (equation 3.2) is to discretize the problem over the feasible region. Let Pki. ,j = 0. 1,. , K(j); ,j = 1, 2,. . . , n, represent given points satisfying XoJ = 0 and 1 < Xkj 6 uj for k # 0. Problem (equation 3.2) can be approximated by the following multiple choice knapsack problem.
j=l
subject
k-l
to
k&;.kj
= I,
j = 1,2 ,...,
ikj = 0 or 1.
n
Capital
budgeting
problems
in pavement
maintenance
179
Sinha and Zoltners [9] present a solution algorithm for solving this problem. Through repeated refinements of the approximation process, or by choosing a very small grid originally, an e-optimal solution for problem (3.2) can be obtained. Armstrong et al. [l] present a branch-and-bound algorithm for solving (equation 3.2) directly. A convex relaxation of the problem is solved at each node and the dichotomy performed restricts xj to be 0 down one branch and to be in [l, Uj] down the other branch. List structures similar to those used successfully by Barr and Ross [3] to solve the zero-one knapsack problem are present in the computer code implementation of the algorithm. This code is available, for academic use, from the authors upon request. An extension of the algorithm from [l] to solve problem (3.3) is currently under study by the authors. However, as yet, no computer code is available. The discretization procedure previously described for problem (3.2) can also be utilized on problem (3.3) to create a multiple choice knapsack problem. Here, we let Xkij, represent
the selected
K(i, j); ,j = 1,2,. . . , n; i = 1, 2, . . . I(j)
k=O,l,..., approximation -Uoij = 0
The problem
points. 1 6 xkij < uij, 1 < k < K(i,,j)
and
becomes: I(j)
Maximize
i j=l
subject
1 i=l
Wi.3
x
f;j(~,j)&.ij
k=l
to
I(j)
1 i=l
Wi,j)
1
E.kij = 1;
j = 1,2,.
. . , n,
k=O
~kij = 0 or 1
5. SAMPLE
PROBLEM
A sample of 400 projects was selected from records of the Ontario Ministry of Transportation. Associated with each project was a capital outlay x and, utilizing the subsequent performance related to each, f(x) was computed. This data was split into 20 categories according to the average daily traffic (ADT) ((2000 ADT, 200&4000 ADT. 4000-8000 ADT and > 8000 ADT) and region of the province (Northern, Northwestern, Eastern, Central and Southwestern). The number of data points varied from region to region. A regression was performed on the data in each of the categories, using a quadratic model which seemed to provide a good fit. The results of the regression are shown in Table 1. These values provide the objective function parameters for the quasi-integer knapsack model problem (3.2). In the particular example solved as an illustration, the total budget available was taken as B = $264,590. The minimum expenditures aj are also given in Table 1. In the example we have taken each of the 20 functions as representing a specific project. This 20 project problem was then solved using the algorithm of [l]. The results are shown in Table 2. In the general case with many projects there would be several in each category. All projects in the same category would be represented by the same quadratic, on a per mile basis. However, the scaling parameter. the minimum expenditure and the value .f(x) (which is a function of length of the road section) would depend on the particular project in that category.
1. Data
-5.0
27.1
25.5
31.8
400& 8000
>8000
-2.1
-3.6
b -3.0
2F.4
9.4
8.3
12.0
G.0
North
-2.84
-3.92
-5.91
d -3.04
28.8
24.8
31.9
3:3
is given from a regional planning model single linear constraint
<2000 2OOtS 4000
Traffic
Table
-4.1
- 3.0
-4.2
b -3.1
9.0
7.0
10.3
28c.o
Northwestern
-4.2
-3.49
-4.79
d -3.35
32.2
36.7
46.1
2:0
-3.8
-4.2
-5.3
b - 2.8 ;.6
9.0
12.0
17.0
Eastern
-4.14
-4.46
-6.27
d 3.33
26.1
33.3
40.3
-2.5
-5.3
-3.8
b ZFY.0 -2.1
F.5
8.7
12.4
12.0
Central
in pavement maintenance for the Province of Ontario. Objective functions has coefficient a (in thousand of dollars). The upper bound on the variables
-2.9
-6.12
-4.58
d -2.17
38.4
19.5
36.7
4r.3
are of the form is 1.1
-2.1
-1.3
-3.6
b -3.9
9.3
4.0
11.0
sc4
Southwestern
the
-2.7
-1.37
-3.9
d -4.55
hx* + cx + d and
S
H
n
fr
;
g
P > E 2 g z
Capital Table Traffic < 2000 200&4000 400&8000 > 8000
budgeting
problems
2. This table gives the optimal North 1.1 0 0 1.09
in pavement
solution
Northwestern 1.1 0 0 0
to the model Eastern 0 1.1 0 0
181
maintenance described Central
I .04 0 0 1.1
in Table
2
Southwestern 0 0 1.0 1.0
6. CONCLUSIONS
The problem of maintaining high quality roads with limited funds is encountered continually by highway authorities throughout the world. As additional roads are constructed and traffic frequency increases, this problem is becoming more complex. Certain steps have been taken to quantify aspects of pavement maintenance. Controlled studies [3, 12, 131 have provided reasonable approaches to measuring the value of a road and estimating deterioration. As the result, data and pavement conditioning rating functions are available as input to mathematical programming models which are designed to budget capital to pavement maintenance projects. Two such models are presented in this paper. It is hoped that one contribution of this paper will be to provide a framework for future modeling research in the area of pavement maintenance. Valuable input to the decision making processes of highway authorities can be derived through a utilization of the knapsack models of this paper. Obvious extensions of models presented in this paper are to consider budgeting in future time periods and to enter restrictions on the number of roads resurfaced in a given subarea. The majority of these model extensions can be constructed as, or at least approximated by, integer programming problems. For certain classes of these problems, such as those presented herein, efficient solution procedures exist. For the more complex problems, one should expect to expend a great deal of computer time to obtain an optimal solution or be satisfied with a suboptimal solution. REFERENCES 1. Armstrong R. D., Cook W. D. and Palacios-Gomez F. An Algorithm for a Quasi-integer Nonlinear Knapsuck Problem. Working Paper, The University of Texas at Austin. 2. Armstrong R. D. and Sinha P. An application of quasi-integer programming to menu planning with variable portion size. Mgmt Sci. 21, 474482 (December 1974). 3. Barr R. S. and Ross G.-T. A Linked Lisr Data Structure f;,r a Binury Knapsack Algorithm. Center for Cybernetic Studies, Report 232 (August 1975). 4. Ciaffey P. .I. Running costs of Motor Vehicles as affected by Road Designs and Trujic. The National Academy of Sciences (1971). 5. Garfinkel R. S. and Nemhauser G. L. Integer Programming, Wiley, New York (1972). 6. Geoffrion A. and Marsten R. E. Integer programming algorithms: a framework and state-of-the-art survey. Mgmt Sci. 18, 465-491 (1972). I. Mangasarian 0. L. Nonlinear Programming, McGraw-Hill, New York (1969). 8. Pelensky E. The Cost of Urban Car Traoel. Ontario Ministry of Transportation and Communications (1970). 9. Sinha P. and Zoltners A. The multiple choice knapsack problem. To appear in Operations Research. 10. “The ASSHO Road Test,” Highway Research Board of the National Academy of Sciences and the National Research Council, Report No. 5, Washington, DC (1962).