Highway improvement project selection by the joint consideration of ...

© 2012 Operational Research Society Ltd. All rights reserved. 0160-5682/12

Journal of the Operational Research Society (2012), 1–13

www.palgrave-journals.com/jors/

Highway improvement project selection by the joint consideration of cost-benefit and risk criteria 1

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P Kelle  , H Schneider , C Raschke and H Shirazi 1

Department of Information Systems and Decision Sciences, Louisiana State University, Baton Rouge, 2 LA, USA; Highway Safety Research Group in the Department of Information Systems and Decision 3 Sciences, Louisiana State University, Baton Rouge, LA, USA; and Louisiana Department of Transportation and Development, Baton Rouge, LA, USA Since highway improvement project selection requires screening thousands of road segments with respect to crashes for further analysis and final project selection, we provide a two-step project selection methodology and describe an application case to demonstrate its advantages. In the first step of the proposed methodology, we will use odds against observing a given crash count, injury count, run-off road count and so on as measures of risk and a multi-criteria pre-selection technique with the objective to decrease the number of prospective improvement locations. In the second step, the final project selection is accomplished based on a composite efficiency measure of estimated cost, benefit and hazard assessment (odds) under budget constraints. To demonstrate the two-step methodology, we will analyze 4 years of accident data at 23 000 locations where the final projects are selected out of several hundred of potential locations. Journal of the Operational Research Society advance online publication, 2 May 2012 doi:10.1057/jors.2012.55 Keywords: highway improvement project; decision support system; public project; transportation safety; multiple criteria selection; data envelopment analysis

Introduction The appropriate allocation of highway safety improvement funds is an important issue due to the large safety and cost consequences. A study by Blincoe et al (2002) showed that the costs of traffic crashes were on the average US$820 per capita in 2000 in the USA. During the same year, $120 billion of federal highway funds were budgeted in the USA of which 49% were allocated for traffic improvements alone (see Blincoe et al, 2002). States individually spend millions of dollars each year on road safety projects. Every year transportation departments have to screen thousands of road segments and preselect a number of them for further evaluation and then, finally, select a small number of road improvement projects to meet the budget constraints. While the pre-selection criteria often include a variety of measures of road safety, a cost-benefit analysis is used to evaluate this smaller number of road segments to make the final selection of projects to be funded. The pre-selection 

Correspondence: P Kelle, Department of Information Systems and Decision Sciences, Louisiana State University, E J Ourso College of Business Admin, 3195 Taylor Hall, Baton Rouge, LA 70803, USA. E-mail: [email protected]

does usually not include any construction costs because of the difficulty of assigning these to thousands of roads segments with as yet unknown necessary countermeasures. Thus the pre-selection focuses mostly on risk assessment. We propose using the odds against observing a mean crash count at or above the mean crash count at a location as a criterion for pre-selection. The higher the odds against observing a certain crash count at a randomly selected road segment, the higher the potential hazard for the road segment. The cost-benefit analyses that are employed to obtain a final project selection usually only consider the costs of a construction project and the benefits in terms of crash cost reduction. Thus, road projects with a larger reduction in the number of crashes are favoured over those with a smaller reduction provided the construction costs are the same. This tends to favour crash locations with high number of crashes to be included in the final selection, regardless of the risk involved. On the one hand, highways with high volume of traffic such as interstates also have a high number of crashes, although these highways are the safest with respect to crash rates per vehicle mile travelled; on the other hand, highways with a low traffic flow may exhibit a higher risk for drivers to be in a crash,

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but these highways may not have enough crashes to be selected through a cost-benefit analysis. Thus, taking into consideration only costs and monetary benefits biases road improvement project selection toward highways with a high volume of traffic. While this seems sensible from a cost perspective, it is unacceptable from a standpoint of safety. The Ford Pinto case serves as an example where a purely cost-benefit analysis had disastrous consequences. The model became the focus of a public outcry when it was alleged that the car’s design allowed its fuel tank to be easily damaged in a rear-end collision which sometimes resulted in deadly fires and explosions. Ford motor allegedly was aware of this design flaw but refused to pay for a redesign because a cost-benefit analysis showed that it would be cheaper to pay off possible lawsuits for resulting deaths. Although the company was acquitted of criminal charges, it lost the confidence of the customers and thereby tainted Ford’s name as well. There are many situations where the public does not accept decisions based solely on a cost-benefit analysis because of the perception of an unacceptable risk. Many public projects such as road safety improvements and public health projects require more careful consideration of other objectives besides cost and benefits. Therefore we propose that a measure of hazard for a road segment be included into the cost-benefit analysis. This paper discusses a two-step approach, one for the pre-selection of road segments for further analysis and another step for the final selection of road improvement projects. Both steps will include other criteria besides costs. The paper is organized as follows: The next section provides an overview of related literature. In the subsequent section we justify the proposed measures used to assess the risk perception (road hazard). The latter section introduces step one for the pre-selection of road segments based on multi-criteria ranking and uses proxy criteria. This step is necessary because it is difficult and time consuming to provide project costs and benefits estimate at thousands of prospective high-hazard or high-benefit sites. In the subsequent section we discuss the second step, the final project selection, which is accomplished based on a composite efficiency measure of estimated cost, benefit, and hazard assessment under budget constraint. The sixth section provides an example using four years of crash data from Louisiana. In the penultimate section we evaluate and compare our proposed approach with other methods. The last section summarizes the remaining problems and future extension possibilities.

Literature review for highway improvement project selection There is a large body of research that deals with costbenefit analyses. We mention here only a few articles that

are directly relevant to our research and concentrate on the cost or benefit of highway projects using various outcome measures. Sinha et al (1981) analysed the reduction in the expected number of accidents due to highway improvements. The effect of improvements on the severity of accidents was considered by Skinner (1985). Sinha and Hu (1985) evaluated the safety impacts of highway projects using various measures and Pal and Sinha (1998) estimated the effectiveness of projects in reducing crashes. An incremental cost–benefit analysis approach toward highway projects was applied by Farid et al (1994). More references can be found in the papers cited. The above-referred articles, like most other research papers, deal with the evaluation of a single project. However, transportation departments have to select a limited number of road improvement projects to fund out of thousands of prospective choices given a fixed budget. To find an optimum selection of projects from a limited set of projects, Melachrinoudis and Kozanidis (2002) applied a mixed integer knapsack solution to project selection maximizing the total reduction in the expected number of accidents under a fixed budget constraint. Most decisions on project selection involve immediate costs while benefits spread over many years into the future. Brown (1980) applied dynamic programming to obtain a set of projects which provide an optimum, taking into consideration not only present costs but also benefits over several years into the future. A life-cycle cost evaluation method to highway safety improvement projects was applied by Madanu et al (2010). Only few articles deal with the selection of public projects using criteria other than costs and monetary benefits. For instance, Norese and Viale (2002) used a multi-criteria sorting procedure to support public decisions; Yedla and Shrestha (2003) discussed a selection method for environmentally sustainable transport systems; Hinloopen et al (2004) applied ordinal and cardinal judgment criteria in the planning of public transport systems; and Odeck (2006) used the Data Envelopment Analysis (DEA) approach for measuring target performance for traffic safety. Tudela et al (2006) compared the cost-benefit analysis with a multicriteria method applied to transportation projects. While all five articles support the need for other than pure cost-benefit analyses they use different approaches. We believe that the DEA in combination with a multi-criteria decision-making approach is most suitable for the problem under consideration in this paper. There is a wealth of literature on multi-criteria decision making (MCDM). In our review of the literature we will concentrate on a few articles which are related road project selections. Recently Ghorbani and Rabbani (2009) published a multi-objective algorithm for project selection problem. Two objective functions have been considered to maximize the total

P Kelle et al—Highway improvement project selection by multi-objectives

expected benefit of selected projects and minimize the variation of allotted resources. Kozanidis (2009) solved a knapsack problem with two objectives: profit and equity. The second objective minimizes the maximum difference between the resource amounts allocated to any two sets of activities. Zongzhi et al (2010) developed a heuristic approach for a system-wide highway project selection to achieve maximal total benefits. Teng et al (2010) published an empirical study of the highway budget allocations in northern Taiwan. Rudzianskaite-Kvaraciejiene et al (2010) evaluated the effectiveness of road investment projects in Lithuania based on several economic, social and environmental criteria. Summarizing the above contributions in public project selection methodology, we observed that some critical issues are missing that are integral parts of many public projects selection, like our highway improvement case. These include a consideration of the following issues: (1) an evaluation of hazard perception, (2) a very large number (several thousand) of potential projects, (3) a pre-selection process based on proxy measures, and (4) a difficult and expensive cost and benefit evaluation. Our project selection method attempts to overcome the above deficiencies. In the next section we describe a method for assessing hazard perceptions.

benchmarks is to develop statistical models for crash counts. To model the number of crashes over many locations with varying ADT, the Negative Binomial distribution has been studied by many researchers. The Negative Binomial distribution is constructed by assuming that the expected value of the Poisson distribution is a random variable described by the gamma distribution. The first application of the Negative Binomial distribution to accident statistics was discussed by Greenwood (1920) and Arbous (1951). More recently, Miaou (1994), Poch and Mannering (1996), and Hauer (1997) applied the negative binomial distribution to crash statistics on roadways. A cutting edge research in crash-count analysis has been published lately by Lord and Mannering (2010). The Negative Binomial distribution assumes that counts of crashes are drawn from a Poisson distribution. The probabilities for yi crashes at road section i is given by PðY ¼ yi jxi Þ ¼

eli Li ðli Li Þyi yi !

ð1Þ

where xi is a vector describing the road characteristics such as ADT and design features of the road, Li is the length of the road section and the mean of the distribution is parameterized as a log linear model:

Measuring hazard perception in road safety Transportation departments often analyse crash data and try to identify the so-called ‘black spot’ crash locations which are road segments with a higher than expected number of crashes for a specific site type. Nevertheless, drivers involved in crashes often have their own risk perception concerning road hazards and file law suits against states. For instance, the State of Louisiana spends on the average 30 million dollars each year settling lawsuits with plaintiffs injured in crashes which supposedly are due to hazardous road segments. Short of finding a road defect, given the large variation of highway types, Average Daily Traffic (ADT), and the design features of highways, it is difficult to find a single measure that reflects risk perception appropriately. In fact, it is doubtful that there is one single criterion that is able to incorporate the wide array of hazard perception of humans based on the available data of crash reports. Some may consider a road to be hazardous if there are too many crashes, or too many injury crashes, or too many fatal crashes, or the crash rate is too high, or the percentage of fatal crashes to all crashes is too high; some safety professionals consider the case of too many run-off road crashes, or too many side impact crashes as a sign for a road section to be hazardous. Clearly, one needs benchmarks to determine what is ‘too many’. One approach of obtaining

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0

ln li ¼ xi b

ð2Þ

where b is a parameter vector describing a linear relationship between the mean and the vector of covariates xi. The subscript i denotes a specific road section, i ¼ 1, . . . N, where N is the total number of road sections used in the analysis. To account for the unobserved heterogeneity between road sections one can specify 0

ln li ¼ xi b þ ln ui

ð3Þ

where ui is a random variable with mean 1 and variance 1/y. Using this, the unconditional (over u) distribution is

f ðyi jxi Þ ¼

Z1

f ðyi jxi ; uÞgðuÞdu

ð4Þ

0

Using a gamma distribution having the shape parameter y as the reciprocal of the scale parameter we obtain the probability distribution as

f ðyi jxi Þ ¼

Z1 0

eli Li ui ðli Li ui Þyi ðyLi ÞyLi uyLi 1 eyLi ui dui ð5Þ yi ! GðyLi Þ

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Solving the integral yields the Negative Binomial distribution for the crash count f ðyi jxi Þ ¼

  yi GðyLi þ yi Þ li Gðyi þ 1ÞGðyLi Þ li þ yLi  yLi yLi  li þ yLi

ð6Þ

variance of either the estimate of the overall mean or the estimate of the individual mean. The posterior distribution derived from prior and the observed data is used to compute tail probabilities for the empirical Bayes estimate (see for instance El-Basyouny and Tarek, 2006). The empirical Bayes estimate is the weighted sum of the mean crash count li and the observed crash count Yi EBi ¼ wi li þ ð1  wi ÞYi

Thus for each site, i, it is assumed that the crash count has a Negative Binomial distribution with parameters depending on ADT, on the section-specific features, and on a common so-called over-dispersion parameter y multiplied by the length of the road section. The negative binomial model has a longer tail than the Poisson distribution depending on the magnitude of the dispersion parameter. Note that li represents the mean number of crashes for a specific time period which may be one year or multiple years. If multiple years are used this will affect the mean and the variance. For the purpose of our methodology, the tail probability of the distribution of the mean serves as a hazard or risk assessment for a specific road segment. For instance, for any given value, c, the fraction of road sections with a mean above c is given by the tail probability of the prior gamma distribution Z1 c

ðyLi =li ÞyLi yyLi 1 eðyLi =li Þy dy ¼ p GðyLi Þ

ð7Þ

The smaller this probability is, the less likely it is to find road segments that have such low probabilities. Instead of using the small tail probabilities, one might prefer the odds (1p)/p against observing a given mean crash count larger than c based on the specified model. Odds can easily be compared regardless of the mean or variance and they have the advantage that non-experts have a fairly good understanding of odds. For instance, if the tail probability for c crashes is 0.001, then the interpretation is that it is 999 times more likely that we will observe a mean crash count below c than for c or above for a randomly selected road segment with identical ADT and engineering features. Odds are widely used in risk applications and thus can be used to associate a hazard or risk with a certain crash count of a road section. The larger the odds are, the larger the hazard or risk is. In most practical situations the true mean for crashes at locations has to be estimated and the issue of confidence intervals has to be addressed. Many authors (see for instance Cheng and Washington, 2005 and El-Basyouny and Tarek, 2006) have discussed the use of the empirical Bayes estimate rather than the average as an estimate for the mean. Morris (1983) showed that the mean squared error for the empirical Bayes estimate is smaller than the

ð8Þ

and its variance is VarðEBi Þ ¼ wi ð1  wi Þli þ ð1  wi Þ2 Yi

ð9Þ

where wi ¼

1 1 þ z2i =s2i

ð10Þ

is the weight given to the overall mean li, and s2i ¼ li Li

ð11Þ

is the within-sample variance and ðli Li Þ2 ð12Þ yLi is the between-sample variance. Then the posterior distribution of the mean crash count is also gamma with parameters depending on the mean and variance of the empirical Bayes estimate. If we denote z2i ¼

bi ¼

EBi VarðEBi Þ

and

ai ¼ bi  EBi

ð13Þ

then the fraction of road segments with a crash count at or exceeding the value c is given by Z1 c

bai i yai 1 ebi y dy Gðai Þ

ð14Þ

In continuing with the description of our approach, we notice that the odds for the number of crashes are only one measure of the potential road hazards. Other measures that are suitable include the odds for injury crashes, run-off road crashes, side impact intersection crashes, etc. The next section will discuss a methodology to incorporate different measures using multiple criteria ranking.

The pre-selection of road segments based on multi-criteria ranking—hazard efficiency measure A highway improvement project is an improvement type applied to a site (road segment). Usually at each site different safety improvement types (such as traffic flow improvement, new pavement, traffic signs, etc) are possible

P Kelle et al—Highway improvement project selection by multi-objectives

with different costs and benefits of accident cost reduction. For benefit estimates there are standard accident modification factors that have been developed by researchers. However, improvement cost estimates are time consuming and require costly engineering analysis; therefore in practice only a small number of sites are considered for improvement. We propose a pre-selection method to identify the prospective improvement sites for cost and benefit estimates chosen from thousands of prospective road segments. The major goal of this pre-selection is not to identify the best sites for safety improvements but to eliminate sites that are not preferable from the point of view of any of the criteria that are important for a particular study. For instance, for rural road segments without intersections, the number (odds) of run-off road crashes may be considered important, while for an intersection study the number (odds) of side-impact crashes is important. Since the costs and benefits of the projects are not readily available and require an in-depth analysis, for the pre-selection we will apply the hazard measures discussed in the previous section. These are proxy measures which are related indirectly to the benefit measures derived from available crash counts of our data. Since several measures of safety hazard may be appropriate, as pointed out above, we will consider multi-criteria ranking to facilitate the selection of the potentially important sites for safety improvement. The goal of our pre-selection method is to (1) eliminate thousands of road segments that do not qualify for consideration of safety improvement, (2) to not eliminate those sites which may be advantageous in any one of the hazard criteria or in any weighted combination of the criteria, (3) and to require minimal user input. Although the literature of multi-criteria ranking and selection methods is very rich, the majority of those methods are not applicable under the above conditions. Non-parametric methods, such as PROMETHEE, SMART, ELECTRE, and the AHP-based methods (see, eg, in Brans et al, 1985, or Salminen et al, 1998) use cross efficiency ranking that requires user input of pairwise comparisons; therefore, they are not applicable for thousands of sites as in our case. Parametric methods, such as weighted scores, are applicable for ranking thousands of projects in theory, but in practice it is difficult to obtain user input on the appropriate weights. We propose the use of DEA which is based on preference weights without user input. DEA was first developed by Farrell (1957) and consolidated by Charnes et al (1978) as a non-parametric procedure that compares decision units using performance indicators. The DEA method has been applied for ranking in several different areas (see, eg, Cooper et al, 1999, which contains a list of over 1500 references). Recent extensions and applications include a matrix-type network DEA algorithm (Amatatsu et al, 2012) and its application for the performance measurement of a transportation network (Zhao et al, 2011).

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In DEA, the preference weights are calculated by linear programming, a method which can easily be applied to evaluate and rank thousands of sites. DEA is an extreme point method, that is, it compares each site to all other sites with weights calculated to be the most favourable for the particular site being evaluated. This is the major advantage for our goal because it ensures that none of those sites which may be advantageous in any one of the important criteria (or in any weighted combination of the criteria) are eliminated. Only those sites are eliminated that are not preferable according to any weighted combination of the criteria, including the most favourable one. In DEA, the efficiency of the decision-making unit (DMU) is the weighted output over weighted input. The objective of the DEA is to identify the DMU that produces the largest values of outputs by consuming the least amount of inputs. For instance, the input would be the cost of the project, and the output would be the gain in safety which can be measured by the crash cost reduction. Let s be the number of different output criteria and m be the number of input criteria used and let N be the number of different sites (DMUs). Let us consider a specific site k ¼ 1, 2, . . . , N, where Rik represents the measure of the ith output criterion (i ¼ 1, 2, . . . , s) and Vjk represents the measure of the jth input criterion ( j ¼ 1, 2, . . . , m) for site k. The efficiency of site k is measured as the weighted sum of outputs over the weighted sum of inputs (as in productivity measures) Ps vik Rik Ekk ¼ Pmi¼1 ð15Þ u j¼1 jk Vjk By using DEA we attempt to find optimal weights, ujk, and vik, for each DMU that maximizes the site efficiency, Ekk, by comparing each site, k, with all other sites subject to the restriction such that the weights are nonnegative and all Eknp1 for n ¼ 1, 2, . . . , N. For each site, k, the above optimization problem can be described as the equivalent linear programming problem Ek ¼ max Ekk Ps vik Rin s:t: Ekn ¼ Pmi¼1 p1 j¼1 ujk Vjn vik X0

ðn ¼ 1; :::; NÞ

ði ¼ 1; :::; sÞ ujk X0

ðj ¼ 1; :::; mÞ

ð16Þ

with Ekk defined in (15). The efficiency of a DMU, the Ek value, is the DEA efficiency measure for site k. There are two basic cases: Ek ¼ 1 site k is efficient (Pareto-optimal site), and Eko1 site k is not efficient; it is dominated by other site(s). To illustrate this concept, consider two output criteria such as the odds for a crash count and the percentage

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of injury crashes at each road segment. For two criteria the idea of DEA can be shown in a graph. In Figure 1, the points A, P and B represent the road segments. The line y2-A-B-x2 is the efficient envelope. Any road segment that has a weighted average of the two criterions inside the envelope is not as efficient as the points on the envelope. The road segment P has an efficiency computed as the distance OP divided by the distance OD. Note that higher efficiency of a site in our context means better selection for safety improvement. Therefore, for the pre-selection we consider only output measures that are to be maximized without any input measures included. Also, rather than considering the hard-to-measure improvement in safety, we use the hazard perception measures described in the previous section, as proxies that are based on readily available data. The objective is to evaluate road segments based on different criteria, such as high odds for mean crash counts, a high percentage of injury crashes, a high percentage of run-off road crashes, etc and combinations of these measures. Based on the DEA method of (16), for each site we calculate the HEk ¼ Ek value. This is considered as the Hazard Efficiency Measure of site k. This measure can be applied when the sites are ranked based on hazard because it provides an efficiency comparison using the distance to the efficient surface. The literature mentions the potential disadvantages in using DEA ranking, such as too many efficient sites, the sensitivity of the selection of the sites included in the list of sites, and the data estimation error. Since the goal of our pre-selection is to rank thousands of road segments and eliminate road segments that are dominated by other road segments the disadvantages of the DEA ranking are of little concern. The main advantage of the proposed Hazard Efficiency Measure based on DEA ranking is that it produces a single hazard score for each road segment and thus

A Criterion Y

y2 D P

B

y1

allows ranking them and selecting a fixed number of topranked sites with the highest hazard scores for further analysis.

Methodology of the project selection under budget constraint considering benefit and risk objectives jointly—composite efficiency measure The above pre-selection (Step 1) provides a set of sites that have the highest potential hazardous conditions dominating the rest of the sites and are thus candidates for an evaluation by traffic engineers. For the pre-selected sites, the following estimates are prepared: K K

expected crash cost reduction (benefit) and project cost estimates.

Both measures depend on two variables, the site and improvement type and require a detailed engineering analysis. Different improvement types are possible at each site which are considered separate projects, although usually only one of these improvements is selected at a site. The final project selection is an MCDM problem of selecting a set of projects with the following objectives: K K

maximum benefits in accident cost reduction, located at maximum risk sites,

subject to a fixed budget constraint for the total cost of the selected projects. We calculate a Composite Efficiency measure, CEp, for each prospective project p, as a composite of estimated cost, benefit, and hazard assessment. In order to avoid the requirement for a significant user input for the weight selection (pairwise comparison or subjective grading), we apply the DEA method (16) as in the previous section for the Hazard Efficiency measure. However, besides the hazard measures we include here the benefit and cost measures creating a Composite Efficiency measure. We use the hazard measures and project benefit as output (to maximize) over the project cost as an input (to minimize). The goal of the final selection is to maximize the composite efficiency scores, CEp, of the selected projects under the available budget, K. Let Cp be the cost for project p, then the MCDM problem is reduced to the following 0–1 Knapsack Model Max Sp xp CEp subject to Sp xp Cp pK; xp ¼ 0 or 1

0

x1

ð17Þ

x2 Criterion X

Figure 1 A DEA illustration.

where each project p is either selected (xp ¼ 1), or not selected (xp ¼ 0).

P Kelle et al—Highway improvement project selection by multi-objectives

The method of combining the DEA ranking with MCDM has been used for different applications. For instance, Golany (1998) combined interactive, multipleobjective linear programming with DEA; Stewart (1996) compared the concepts of efficiency and Pareto Optimality in DEA and MCDM. Furthermore, Belton and Stewart (1999) stated that MCDM is generally applied to ex-ante problem areas where data are not readily available such as in the case of future technologies. DEA, on the other hand, provides an ex post analysis of the past as a basis to learn. Since our project selection case is based primarily on existing statistical data related to such areas as accident risk, cost, and benefits, the DEA ranking gives valuable information for selecting the future improvement projects. We believe that the advantage of our methodology is that it can be applied to project selections in which (1) tens of projects are to be selected out of hundreds of potential projects pre-selected from thousands of potential sites and (2) the selection is based on multi-criteria objectives which require no user input information regarding the weights or pairwise comparison of the projects. We illustrate the two-step procedure through an application in the following section.

Case discussion and analysis Our case study is based on four years of accident data from Louisiana comprising over 160 000 crashes per year at more than 23 000 locations with estimated crash costs close to $6.5 billion. The data base we are using is the result of a large project between Louisiana Highway Safety Commission (LHSC), the Department of Transportation and Development (DOTD) and the Highway Safety Research Group (HSRG) at Louisiana State University. The statistical data are based on information obtained on traffic crashes submitted by state, sheriff, and local police agencies throughout the state of Louisiana. We demonstrate the use of our procedure with a specific example of 5091 rural two-lane two-way roadway segments without intersections.

the section ‘Measuring hazard perception in road safety’ using four years of Louisiana crash data. The covariates included ADT, lane width, and shoulder width. Other variables such as shoulder type, roadside hazard rating, driveway density, horizontal curvature, vertical curvature, centreline rumble stripes, passing lanes, lighting, and grade level could have been considered but were not readily available. This lack of availability of complete information on each road segment does not in any way affect the methodology as a whole; however, the more covariates are available the smaller is the unexplained variation in the regression model. Once the pre-selection of road segments has been done, for a manageable number of sites the additional road design features can be easily provided for the final selection. We also want to point out that this case serves only as an example to demonstrate the methodology. Measuring the hazard perception, we use the odds against observing the specific average number of crashes for each site. A Negative Binomial model was fitted for the base condition with a shoulder width of six feet and a pavement width of 24 feet for the three sets of crash counts (number of total crashes, injury crashes and run-off-road crashes). Next the accident modification factors published in the Highway Safety Manual were applied for the 5091 locations to obtain predicted crash counts. The ratio of total actual crash counts and total predicted crash counts was used to adjust the parameter bo to better reflect the level at all 5091 locations. Based on these three models, the probability (and the associated odds) of the mean crash count exceeding the estimated crash count was computed for each road segment. We derived the following three hazard measures based on readily available data: K K K

To prepare the hazard measure (odds) for the pre-selection, a Negative Binomial regression model is fitted according to

R1: the odds against total crashes, R2: the odds against injury crashes, R3: the odds against run-off-road crashes.

All three measures are of interest for rural highways without intersections. The estimates for each of the three models are displayed in Table 1. A fourth measure was added for further examination: K

Hazard efficiency measures

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R4: the total crash cost for site k.

Note that those sites which have the worst hazard measures (higher odds) and higher crash costs are preferable for

Table 1 Estimates for negative binomial models Model All crashes Injury crashes ROR crashes Standard errors are in brackets.

Intercept b0

b1

ln(y)

6.1213 (0.982) 8.6568 (1.258) 6.5884 (1.2861)

0.74698 (0.109) 0.92685 (0.152) 0.67074 (0.1563)

0.063 (0.018) 0.01157 (0.0233) 0.010006 (0.0205)

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Table 2 An overlap of road segment selection results for different selection criteria Criterion

R1 R2 R3 R4 DEA 1–3 DEA 1–4 (%) (%) (%) (%) (%) (%)

R1 R2 R3 R4 DEA 1–4 DEA 1–3

100 68 63 52 56 78

68 100 60 57 60 73

63 60 100 48 51 78

52 57 48 100 93 56

56 60 51 93 100 62

78 73 78 56 62 100

improvement selection; thus R1-R4 are all output measures to maximize in (16). We pre-selected 5% of the 23 000 sites using different combinations of the above four criteria (R1-R4) and compared the different sets of road segments obtained by using different criteria. Table 2 shows the percentage of overlap when using different ranking criteria. The pre-selected sites arranged by their score according to the odds R1, R2, and R3, the crash cost, R4, and the DEA 1–4 score (using all four criteria, R1-R4) and the DEA 1–3 score (using just three criteria, R1-R3). Using R1, R2, or R3, individually, results in over 70% of the selected sites overlapping with the DEA 1–3 score (using R1, R2 and R3 combined). The DEA 1–4 score (using R1, R2, R3 and R4 combined) has more overlap (90%) with R4 than with R1, R2 or R3 (which ranges from 50 to 60%). We note that we will not use the crash cost R4 in Step 2 but rather R5, which measures the crash costs reduction. Federal guidelines require that states list at least 5% of the most hazardous road segments. Federal requirements (Sections 148(c)(1)(D) and 148(g)(3)(A), of Title 23, United States Code) stipulate that each state describe at least 5% of its locations currently exhibiting the most severe highway safety needs. The number of hazardous pre-selected sites may be larger for our project selection method depending on how many engineering work hours are available to analyse the pre-selected sites. This selection should contain road segments that are preferable to other sites to be considered for improvement. However, because of limited funds, cost and benefits need to be taken into consideration in the final project selection.

Composite efficiency measure For the final project selection (Step 2), the following estimates are prepared for safety improvement projects at the pre-selected sites: K

R5: the expected crash cost reduction (benefit) of a project that depends on two variables, the site and improvement type.

K

V1: the cost estimate of a project that is also dependent on site and improvement type, requiring a detailed engineering analysis.

Note that R5 is an output measure to be maximized and V1 is an input measure to be minimized in (16). At each site, different improvement types are possible which are considered as separate projects, but usually only one of these improvements is selected at a site. The Highway Safety Manual provides accident modification factors that have been used to predict the expected reduction rate in accidents as a consequence of the improvement type. For simplicity, in our example we use only two possible actions, namely widening the road to a standard width of 24 feet and widening the shoulder to 6 feet. The cost estimate of these actions is based on a cost of $2.5 million per mile plus a proportional cost for the increase in the lane width and the shoulder width of the road segment. Although this estimate is a simplification, it is suitable for demonstrating the methodology. Obtaining an exact estimate of the costs for improvements is not feasible for our example. The new ranking which takes into account the hazard measures (R1, R2, R3) and benefits R5 has about 46 to 55% of road segments overlapping with the set obtained using the individual ranking R1, R2, R3, and 58% with the set obtained through the ranking of DEA 1–4 score and 53% with the set based on ranking the DEA 1–3 score. We applied the zero-one Knapsack Model (17) to select the improvement projects by maximizing the Composite Efficiency measure based on the DEA scores of the projects (CEp, based on the criteria of cost, benefit, and hazard) subject to the budget constraint. For planning purposes, different geographical areas and budget alternatives have been evaluated which required an efficient ‘what-if’ analysis, trade-off curves, and other practical decision support tools. By applying Boolean constraints (as and/or selections), it is easier to consider sites with multiple project options where not more than one option can be selected. Also groups of joint projects can be handled in which either all or none are to be selected. We developed a PC-based computerized decision support system which can easily be accessed in Excel. The input data including the crash information along with other site characteristics which may need user input are also available in Excel. The algorithmic part is run by VBA macros, which support a ‘what-if’ analysis based on different budgets, on subsets of sites to be considered, and which allows different pre-selection criteria and selection limits. The output possibilities are also easy to modify using Excel. The imbedded LINGO (Extended Version 3.01, 1999, LINDO System Inc.) procedures effectively handle the large problems which need to be solved such as the dual LP for DEA evaluating up to thousands of projects in solving (16) and the 0–1 knapsack problem solution for

P Kelle et al—Highway improvement project selection by multi-objectives

several hundred pre-selected projects, using a branch and bound procedure in (17). Alternatively, Excel Solver can also be used, but it requires a longer computation time. In the section ‘Composite efficiency measure’ we discussed how the ranking is affected by using the different measures R1, R2, R3 and R5. In the following section we compare the proposed composite efficiency method for project selection to a pure cost-benefit analysis approach and a pure hazard approach and demonstrate some of the advantages of the composite selection method.

Comparison with other project selection methods Our Composite method jointly considers benefit and hazard objectives expressed in the project Composite Efficiency score which is the weighted average of benefit, hazard, and cost objectives. The weights are calculated with the DEA method (16). For comparison, we considered two alternative selection methods with a single objective each: K

K

Benefit method: selecting the projects with the highest benefit and Hazard method: selecting the projects on the sites with the highest hazard.

In summary, the Benefit method is biased toward including fewer large projects with large benefits while

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the Hazard method is biased toward including smaller projects with higher risk road segments. Overall, the Composite method provides a good compromise between obtaining a large benefit in crash cost reduction and including more projects with high-risk road segments. The results of the three methods are illustrated in the columns of Table 3 and Table 4. In Table 3 we illustrate the results for three different budgets in millions, $200, $400, and $600 million, while in Table 4 we summarize the statistics (average, standard deviation, and percent differences) for a large number of runs with different number of pre-selected sites and a large set of different budgets. The tables show the result of selecting the highest objective value under a fixed budget constraint solving the optimization problem (17). The three different objectives are displayed in different columns, respectively. K

K

The Benefit method with the crash cost reduction as single objective (denoted by R5 in Section ‘Methodology of the project selection under budget constraint considering benefit and risk objectives jointly—composite efficiency measure’); The Composite method with highest Composite Efficiency scores (described in Section ‘Methodology of the project selection under budget constraint considering benefit and risk objectives jointly—composite efficiency measure’);

Table 3 A comparison of the project selection methods for different budget constraints 200

Budget ($million) Objective

600

400

Benefit

Composite

Hazard

Benefit

Composite

Hazard

Benefit

Composite

Hazard

11 NA 7.1 NA

17 55% 3.6 49.6%

16 45% 2.9 59.8%

22 NA 10.8 NA

27 23%* 8.3 23.3%

27 23% 5.2 52.1%

30 NA 12.8 NA

36 20% 10.9 14.6%

37 23% 9.1 28.8%

23.8 64% 8.3 101%

19.2 33% 5.8 40%

14.5 NA 4.1 NA

23.8 64% 8.4 183%

14.5 0% 2.5 16%

14.5 NA 3.0 NA

Remaining R2 hazard (odds of injury crashes, 1 to million): Maximum R2 111.1 111.1 37.0 111.1 Hazard increase 200% 200% NA 267% Average R2 11.2 11.1 7.8 11.3 Hazard increase 43% 42% NA 60%

30.3 0% 7.0 2%

30.3 NA 7.1 NA

111.1 400% 13.1 196%

30.3 36% 6.8 54%

22.2 NA 4.4 NA

Remaining R3 hazard (odds of run-of-road crashes, 1 to million): Maximum R3 58.8 32.3 32.3 58.8 Hazard increase 82% 0% NA 129% Average R3 15.9 12.4 12.3 17.6 Hazard increase 29% 0% NA 64%

31.3 22% 11.3 5%

25.6 NA 10.8 NA

58.8 224% 19.9 79%

25.6 41% 10.9 3%

18.2 NA 11.2 NA

# Projects selected Increase Benefit in $million Loss in benefit

Remaining R1 hazard (odds of total crashes, 1 to million): Maximum R1 23.8 19.2 19.2 Hazard increase 24% 0% NA Average R1 7.7 5.4 5.6 Hazard increase 36% 4% NA

Percentage increase above baseline denoted by NA. Bold entries show the example explained in text.

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Table 4 A summary of comparisons for the different project selection methods Objective

# Projects selected Benefit in $million Remaining max R1 Remaining avrg. R1 Remaining max R2 Remaining avrg. R2 Remaining max R3 Remaining avrg. R3

Overall average

Standard deviation

Average loss % (compared with the best)

Benefit

Composite

Hazard

Benefit

Composite

Hazard

Benefit (%)

Composite (%)

Hazard (%)

23.67 10.91 23.81 8.46 111.11 12.16 58.82 17.88

29.00 8.36 15.90 4.34 42.00 7.07 28.47 11.08

29.00 6.60 14.57 3.64 29.85 6.60 25.19 11.14

9.44 2.38 0.00 0.71 0.00 0.95 0.00 1.59

8.67 3.22 4.22 1.70 34.00 2.13 5.62 1.17

9.12 3.02 5.03 1.54 6.63 1.65 6.09 1.33

18.4 0.0 63.4* 132.5 272.2 84.3 133.5 60.5

0.0 23.3 9.1 19.2 40.7 7.2 13.0 0.6

0.0 39.5 0.0 0.0 0.0 0.0 0.0 0.0

*Bold entries are discussed in the text.

300.0% Percent increase in:

The Hazard method with highest Hazard Efficiency scores (described in Section ‘The pre-selection of road segments based on multi-criteria ranking—hazard efficiency measure’).

250.0%

Although this is a case study limited to road projects in Louisiana, the analysis suggests several general trends. (1) The number of projects selected (first row in Tables 3 and 4). The Benefit method is biased toward larger projects and thus selects fewer projects for a given budget. The Composite and Hazard methods select a larger number of projects. For instance, for a budget of $200 million, the Benefit method selects 11 projects while the Composite method selects 17 projects and the Hazard method selects 16 projects (see in Table 3). On the average, 18.4% fewer projects are selected by the Benefit method than with the other methods (see in Table 4).

50.0%

K

(2) The benefits in millions of dollars (third row in Tables 3 and 4). The Benefit method provides the largest benefit and the Hazard method the lowest benefit as expected. For instance, for a budget of $600 million, the benefit for the 37 projects selected by the Benefit method is $12.8 million while the Composite method provides a benefit of $10.9 million for the 36 projects and the Hazard method provides a benefit of $9.1 million for the 37 projects (see in Table 3). The benefit loss of the Hazard method is 39.5% on the average (see Table 4). The Composite method provides benefits in between the two (23.3% average loss in benefit). However, a major hazard reduction is the trade-off for benefit loss as we see in Figures 2 and 3.

Benefit loss 200.0%

max hazard R2 max hazard R3

100.0%

0.0% Benefit

Composite

Hazard

Figure 2 Risk/benefit trade-off for different project selection methods (with maximum hazard comparison). Percent increase in: Benefit loss

140.0% 120.0%

avrg. hazard R1

100.0%

avrg. hazard R2

80.0%

avrg. hazard R3

60.0% 40.0% 20.0% 0.0% Method: -20.0%

Benefit

Composite

Hazard

Figure 3 Risk/benefit trade-off for different project selection methods (with average hazard comparison).

The hazard at the unselected sites represents the remaining hazard after the selected sites is improved. The hazard is evaluated using two different measures K K

maximum hazard at the unselected sites (worst case) and average hazard at the unselected sites.

In our case, we consider the following hazard types: K

(3) The remaining hazard (starting in row five in Tables 3 and 4).

max hazard R1

150.0%

K K

R1: the odds against total crashes, R2: the odds against injury crashes, R3: the odds against run-off-road crashes.

P Kelle et al—Highway improvement project selection by multi-objectives

For the measures of hazard given above, the hazard reduction is between 33 and 62% for the Composite method and 38 to 73% for the Hazard method (see Figures 2 and 3). The hazard evaluation requires a more detailed explanation. Each hazard in Tables 3 and 4 is expressed as the odds of one million to 1. For instance, at a given location, if the odds are 2 million:1, then the odds are 2 million to 1 against this specific average number of crashes occurring if a road section with the specific site features had been chosen at random. We complete a detailed comparison for all three hazard measures for each selection method. The Benefit method leaves many high risk sites (higher odds) out of the selection. The Composite method results in odds that are in between the Benefit and Hazard methods. As illustrated in Table 3, the hazard increase is different for the three different hazard measures R1, R2, and R3 considering both the maximum and the average odds for the unselected sites. For instance, for a budget of $400 million, the maximum R1 which is not selected with the Hazard method is 14.5:1 (odds of mean crashes is 14.5 million to 1), with the Composite method it is 19.2:1, and with the Benefit method it is 23.8:1. For this case the hazard increase is 33% for the Composite method and 64% for the Benefit method, compared with the best, namely the Hazard method. Using the average rather than the maximum odds R1 of projects not selected the increase is 40% for the Composite method and 101% for the Benefit method. Table 3 also shows the percentage increases in the R2 and R3 odds for the three methods. While the Hazard method results in a 52.1% loss in benefits, the Composite method results in a 23.3% loss of benefits compared with the Benefit method. The Hazard method selects the sites with the highest hazard efficiency score (the weighted average of odds R1, R2, and R3 in our case); therefore we expect that the Hazard method selects the sites with the highest odds for each hazard measure R1, R2, and R3, respectively. However, there are cases where the Composite method provides lower hazard in one of the measures (indicated by a negative hazard increase in Table 3) but still a higher hazard in the other measures. The statistical results based on a large number of runs are summarized in Table 4. This table contains the overall averages, the standard deviations, and the average loss percent compared with the best solution. For instance, when using the Benefit method there is a 63.4% increase in the maximum of R1 of the site not selected and a 132% increase in the average R1 for sites not selected. In contrast to the Benefit method when using the Composite method the hazard increase is only 9.1 and 19.2%, respectively. Similar results are obtained for R2 and R3. The increase in the maximum hazard of sites not selected is very high for the Benefit method and slightly higher for the Composite method when compared with the Hazard method. For

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instance, the increase in the maximum hazard for the Benefit method is 27.2 and 133.5% for R2 and R3, respectively. These increases are only 40.7 and 13.0% for the Composite method. Figures 2 and 3 depict the percentage loss for the maximum hazard and the average hazard criteria, respectively. The figures show the loss in benefits as well as the loss in maximum and average hazard for not selected sites. It demonstrates that the Composite method is a compromise between the two extreme methods, that is, using only cost/benefits or using only hazard.

Discussion and extensions This paper is dealing with the selection of safety improvements projects allocating the fund that is federally mandated for safety improvements. The major goal is to decrease the accident rate using the available budget. This leads to the practice of applying cost-benefit criteria for the safety improvement project selection. From an economic point of view it can be argued that a pure cost-benefit analysis should be used, but this criterion may leave a number of high-risk sites unselected where the traffic is small. Examples such as the Pinto and the recent Shell Oil well disaster cases show that the pure cost-benefit considerations are sometimes unacceptable for the public. For this reason, government and industry often sets limits on risk for products and services and they try to find the best compromise between benefit and risk. We suggest a methodology that includes risk measures combined with benefit and cost measures in the safety improvement project selection process. The numerical example of a case demonstrates the advantages of our multi-criteria decision support method. We propose a new hazard estimation method using the tail probability of the Negative Binomial distribution to express the odds against observing a given crash count at a specific road segment. These odds and other readily available crash statistics are used to pre-select prospective sites for road improvements in the first step, eliminating thousands of sites. For a reasonable number of pre-selected sites a detailed project evaluation can be prepared by the engineers and safety experts. For the multi-criteria ranking we used the DEA to create a single Hazard Efficiency score for each site. This method can be used to rank thousands of sites since it requires minimal user input and it is computationally efficient. The DEA method insures that those sites are not eliminated which may be advantageous in any one of the criterion or in any combination of the criteria. For the pre-selected sites the detailed cost and benefit evaluation results and the hazard scores were combined to create a single Composite Efficiency score for each prospective project used in the final project selection. Overall, the project selection method based on the above Composite Efficiency score provides a good compromise between

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obtaining a large benefit in crash cost reduction and including more projects on high risk road segments. To summarize, our method can resolve the following issues that were not adequately handled in previous methods published in literature: (1) the evaluation of hazard perception, (2) the very large number (several thousand) of potential sites, (3) the pre-selection process based on readily available proxy measures, (4) a final project selection based on the composite measure of cost, benefits and risk. There are some possible extensions to our approach. For instance, one of the inputs requires the estimation of project cost and the estimates of the monetary benefit of an improvement project. However, providing accurate estimates for hundreds of projects may not be feasible. Thus approximate cost estimates as computed for our example may have to be used. The resulting uncertainty motivates the stochastic extensions for the project selection. Therefore, a chance constrained DEA seems to be a possible extension to apply in the future for traffic project ranking and selection. Some recent publications applying this methodology include Cooper et al (2004), Srinivas et al (2006), and Lahdelma and Salminen (2006). Nevertheless, the extensions for fuzzy DEA methodology may also be considered (see eg in Chiou et al, 2005 and Dimova et al, 2006). Other possibilities include the application of the stochastic dominance approach as in Nowak (2007) and the combination of simulation and optimization suggested recently in Scott et al (2007). The major difficulty that arises with these methods is the large number of projects to be considered. Another direction of extensions of our approach includes more user input for the final step where the number of projects to consider is smaller. An extension of the DEA ranking method is the integration of criteria preference scores provided by the users such as, for example, the use of assurance regions (upper and lower bounds for each criterion) for the restriction of weights in DEA which is an approach to include user preferences (see Thompson et al, 1990 and Wong and Beasley, 1990).

References Amatatsu H, Ueda T and Amatatsu Y (2012). Efficiency and returns-to-scale of local governments. Journal of the Operational Research Society 63(3): 299–305. Arbous AG (1951). Accident statistics and the concept of accident proneness. Biometrics 7(4): 340–432. Belton V and Stewart TJ (1999). DEA and MCDA: Competing complementary approaches? In: Meskens N and Roubens M (eds). Advances in Decision Analysis. Kluwer: Dordrecht, MA, pp 87–104. Blincoe L, Seay A, Zaloshnja E, Miller T, Romano E, Luchter S and Spicer R (2002). The economic impact of motor vehicle crashes 2000, U.S. department of transportation. DOT HS 809446.

Brans JP, Mareschal B and Vincke P (1985). A preference ranking organization method (The PROMETHEE method for multiple criteria decision-making). Management Science 31(6): 647–656. Brown DB (1980). Use of dynamic programming in optimally allocating funds for highway safety improvements. Transportation Planning and Technology 6(2): 131–138. Charnes A, Cooper WW and Rhodes E (1978). Measuring the efficiency of decision-making units. European Journal of Operational Research 2(6): 429–435. Cheng W and Washington SP (2005). Experimental evaluation of hotspot identification methods. Accident Analysis and Prevention 37(5): 870–881. Chiou HK, Tzeng GH and Cheng DC (2005). Evaluating sustainable fishing development strategies using fuzzy MCDM approach. Omega 33: 223–234. Cooper WW, Seiford LM and Tone K (1999). Data Envelopment Analysis, A Comprehensive Text with Models, Applications, References and DEA-Solver Software. Kluwer: Dordrecht, MA. Cooper WW, Deng H, Huang Z and Li SX (2004). Chance constrained programming approaches to congestion in stochastic data envelopment analysis. European Journal of Operational Research 155(2): 487–501. Dimova L, Sevastianov P and Sevastianov D (2006). MCDM in a fuzzy setting: Investment projects assessment application. International Journal of Production Economics 100(1): 10–29. El-Basyouny K and Tarek ST (2006). Comparison of two negative binomial regression techniques in developing accident prediction models. Transportation Research Record 1950: 9–16. Farid F, Johnston DW, Laverde MA and Chen CJ (1994). Application of incremental benefit cost analysis for optimal budget allocation to maintenance, rehabilitation and replacement of bridges. Transportation Research Record 1442: 88–100. Farrell MJ (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society, Series A 120(3): 253–290. Ghorbani S and Rabbani M (2009). A new multi-objective algorithm for a project selection problem. Advances in Engineering Software 40(1): 9–14. Golany B (1998). An interactive MOLP procedure for the extension of DEA to effectiveness analysis. Journal of the Operational Research Society 39(8): 725–734. Greenwood MY (1920). An inquiry into the nature of frequency distributions of multiple happenings, with particular reference to the occurrence of multiple attacks of disease or repeated accidents. Journal of the Royal Statistical Society Series A 83: 255–279. Hauer E (1997). Observational Before-After Studies in Road Safety. Pergamon Press, Elsevier Science: Oxford, UK. Hinloopen E, Nijkamp P and Rietveld P (2004). Integration of ordinal and cardinal information in multi-criteria ranking with imperfect compensation. European Journal of Operational Research 158(2): 317–338. Kozanidis G (2009). Solving the linear multiple choice knapsack problem with two objectives: Profit and equity. Computational Optimization and Applications 43(2): 261–294. Lahdelma R and Salminen P (2006). Stochastic multicriteria acceptability analysis using the data envelopment model. European Journal of Operational Research 170(1): 241–252. LINGO (1999). User’s Guide. LINDO Systems, Inc: Chicago, IL, http://www.lindo.com/. Lord D and Mannering F (2010). The statistical analysis of crashfrequency data: A review and assessment of methodological alternatives. Transportation Research Part A 44(5): 291–305. Madanu S, Li Z and Abbas M (2010). Life-cycle cost analysis of highway intersection safety hardware improvements. Journal of Transportation Engineering 136(2): 129–140.

P Kelle et al—Highway improvement project selection by multi-objectives

Melachrinoudis E and Kozanidis G (2002). A mixed integer knapsack model for allocating funds to highway safety improvements. Transportation Research Part A 36(9): 789–803. Miaou SP (1994). The relationship between truck accidents and geometric design of road sections: Poisson versus negative binomial regressions. Accident Analysis and Prevention 26(4): 471–482. Morris K (1983). Parametric empirical Bayes inference: Theory and applications. Journal of the American Statistical Association 78(381): 47–55. Norese MF and Viale S (2002). A multi-profile sorting procedure in the public administration. European Journal of Operational Research 138(2): 365–379. Nowak M (2007). Aspiration level approach in stochastic MCDM problems. European Journal of Operational Research 177(3): 1626–1640. Odeck J (2006). Identifying traffic safety best practice: an application of DEA and Malmquist indices. Omega 34: 28–40. Pal R and Sinha KC (1998). Optimization approach to highway safety improvement programming. Transportation Research Record 1640: 1–9. Poch M and Mannering F (1996). Negative binomial analysis of intersection accident frequency. Journal of Transportation Engineering 122(2): 105–113. Rudzianskaite-Kvaraciejiene R, Apanavicine R and Butauskas A (2010). Evaluation of road investment project effectiveness. Engineering Economics 21(4): 368–376. Salminen P, Hokkanen J and Lahdelma R (1998). Comparing multicriteria methods in the context of environmental problems. European Journal of Operational Research 104(3): 485–496. Scott LR, Harmonosky CM and Traband MT (2007). A simulation optimization method that considers uncertainty and multiple performance measures. European Journal of Operational Research 181(1): 315–330. Sinha KC and Hu K (1985). Assessment of safety impacts of highway projects. In: Carney JF (ed) Proceedings of the Conference on the Effectiveness of Highway Safety Improvements. Nashville, TN, pp 31–40. Sinha KC, Kaji T and Liu CC (1981). Optimal allocation of funds for highway safety improvement projects. Transportation Research Record 808: 24–30.

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Skinner RE (1985). RRR design standards: Cost effectiveness issues. In: Carney JF (ed) Proceedings of the Conference on the Effectiveness of Highway Safety Improvements. Nashville, TN, pp 41–50. Srinivas T, Narasimhan R and Nair A (2006). Vendor performance with supply risk: A chance-constrained DEA approach. International Journal of Production Economics 100(2): 212–222. Stewart TJ (1996). Relationships between data envelopment analysis and multicriteria decision analysis. Journal of the Operational Research Society 47(5): 654–665. Teng J-Y, Huang WC and Lin MC (2010). Systematic budget allocation for transportation construction projects: A case in Taiwan. Transportation 37(2): 331–361. Thompson RG, Langemeier LN, Lee CT and Thrall RM (1990). The role of multiplier bounds in composite analysis with application to Kansas farming. Journal of Econometrics 46(2): 93–108. Tudela A, Akiki N and Cisternas R (2006). Comparing the output of cost benefit and multi-criteria analysis: An application to urban transport investments. Transportation Research Part A 40(5): 414–423. Wong Y-HB and Beasley JE (1990). Restricting weight flexibility in data envelopment analysis. Journal of the Operational Research Society 41(9): 829–835. Yedla S and Shrestha RM (2003). Multi-criteria approach for the selection of alternative options for environmentally sustainable transport system in Delhi. Transportation Research Part A 37(8): 717–729. Zhao Y, Triantis K, Murray-Tuite P and Edara P (2011). Performance measurement of a transportation network with a downtown space reservation system: A network-DEA approach. Transportation Research Part E: Logistics and Transportation Review 47(6): 1140–1159. Zongzhi L, Madanu S, Zhou B, Wang Y and Abbas M (2010). A heuristic approach for selecting highway investment alternatives. Computer-Aided Civil and Infrastructure Engineering 25(6): 427–439.

Received December 2011; accepted March 2012