TWO-STAGE DATA ENVELOPMENT ANALYSIS METHOD FOR TRANSPORTATION INFRASTRUCTURE MAINTENANCE MANAGEMENT

Transcription

1 0 TWO-STAGE DATA ENVELOPMENT ANALYSIS METHOD FOR TRANSPORTATION INFRASTRUCTURE MAINTENANCE MANAGEMENT Emil Juni (Corresponding Author) Graduate Assistant Department of Civil and Environmental Engineering University of Wisconsin-Madison Engineering Drive Madison, WI 0 Phone: erjuni@wisc.edu Teresa M. Adams, Ph.D. Professor, Civil and Environmental Engineering University of Wisconsin-Madison Engineering Drive Madison, WI 0 Phone: 0-- Fax: 0-- teresa.adams@wisc.edu Word count: () + ( x 0 tables/figures) = words November, 0

2 Juni and Adams 0 ABSTRACT Maintenance administrators of transportation infrastructure assets are always facing the challenges of ensuring productive spending and determining the most efficient ways to spend the limited amount of funds and resources available. Efficient resource allocation depends on how efficient the process of maintenance is across all levels of jurisdiction. Especially in states implementing performance-based maintenance management, the knowledge of how efficient a maintenance unit is performing is very important in the state agencies maintenance investment strategy. A Data Envelopment Analysis (DEA)-based framework allows for efficiency comparison of maintenance units. Implementation of this framework in a decentralized fashion allows for comparison of maintenance units within a specific jurisdictional area, providing results showing the particular time period when maintenance units performed efficiently and inefficiently. However, the non-parametric nature of the DEA method limits the framework to compare efficiencies of units included in a particular comparison model. This study introduces the use of two-stage DEA method to improve the existing DEA framework for efficiency measurement of infrastructure maintenance. This involves regressing the values of the efficiency factors and testing their statistical viability before re-running the values through the DEA model. The inclusion of statistical aspects into DEA framework helps to identify factors with significant effect on efficiencies of maintenance units. It also increases the validity of efficiency numbers resulted from the efficiency comparison. Keywords: Asset Management, Maintenance Management, Performance-based Maintenance, Data Envelopment Analysis, Maintenance Investment Strategy

3 Juni and Adams BACKGROUND Assessment of Performance-based Maintenance Performance-based maintenance of transportation infrastructure assets has been an active research topic for many years. Early efforts related to performance-based maintenance for transportation infrastructure started to be noticeable in, when Cabana, et. al. discusses how to address inefficiency and accountability issues in maintenance of low volume roads with performance-based rehabilitation (). In the same year, a guideline for performance measures in highway maintenance operations was developed, as state transportation agencies were beginning to consider changing the way they manage highway maintenance operations, including implementing performance-based system in planning and managing their maintenance programs (). Beginning from these earlier studies, the interest towards learning more about performance-based maintenance of transportation infrastructure assets and improving the efficiency of these maintenance operations continued in the following decade. Through the years, discussions about performance-based analysis were mostly focused on the concept of defining performance measures, target settings, and performance-based resource allocation ()(), and development of statistical models to optimize life-cycle costing and performance of the transportation infrastructure assets ()(). More recently, efforts related to performance-based maintenance have been focusing on the supporting tools or methods in improving the implementation of performance-based maintenance ()(). Condition assessment, performance measures, levels of service (LOS) definitions and thresholds, and the way to incorporate these measures into existing, modified, or new maintenance management systems, are all the common challenges in this subject. The increased understanding of performance-based concepts and methods have been changing the way highway maintenance and operations programs are planned and managed, promoting more objective information on highway condition, greater emphasis on outcome-based performance measures, more emphasis on road-user expectation, more proactive maintenance planning, and implementation of performance-based measures in prioritization and budgeting (). A well-designed performance measurement system evaluate a range of solutions addressing the specific maintenance needs, make trade-offs across different resource allocation options, communicate the implications of different investment levels, and establish targets for results to be achieved based on available resources (). This system of performance measurement and management provides a way for maintenance administrators to evaluate prior work and offer the means to adjust their maintenance plan for the future based on prior performance. Different methods exist in implementing performance-based maintenance management. Case studies in the literature are usually developed for performance measurement done at the statewide level (). Most of them are not looking at individual maintenance units, take into account efficiency concept, or aim to improve efficiency of maintenance units by ensuring the least amount of resources while producing optimal results. This is despite the fact that efficiency of maintenance units is an important aspect in evaluating performance, as it can show how much improvement a unit can get under different conditions. Evaluating efficiencies of individual maintenance unit allows maintenance administrators to improve their overall maintenance investment strategy. One existing framework currently available to assess efficiency of maintenance performance was developed using Data Envelopment Analysis (DEA) method in 00 ().

4 Juni and Adams DEA is defined as a data-oriented approach for evaluating the performance of a set of peer entities called Decision-Making Units (DMUs), and compares and ranks their efficiencies (). In transportation infrastructure, the maintenance units are the DMUs, and the method assesses the performances of these maintenance units in terms of their efficiencies. This makes DEA a suitable approach to use as a tool to implement performance-based resource analysis, which is defined as the way to balance competing objectives and perspectives of multiple stakeholder groups while considering the amount of resources available (). This existing DEA framework specifically focuses on comparing the relative efficiencies of peer maintenance units, and identifies how efficient maintenance units are compared to each other. As DEA is non-parametric, this method is considered deterministic, does not have statistical basis, and produces results with no measure of uncertainty (). In the implementations of DEA method to assess maintenance performance ( ), no specific process was followed to select factors affecting efficiency, which are used as parameters in the DEA models. Efficiency factors are also never statistically validated whether or not they actually have significant effects on maintenance efficiency. This is a big concern, as the result of a DEA modeling is highly dependent upon the efficiency factors used (). Furthermore, interpreting the results from this method simply means comparing efficiency numbers between maintenance units. The efficiency numbers themselves do not mean anything on their own and is only meaningful within the context of comparison with other units in the model. This paper expands on the existing DEA-based framework by implementing a way to ensure that all the efficiency factors selected are significant, while introducing a statistically valid method to measure and compare efficiency of maintenance units. The extra step taken in the DEA modeling helps confirm or deny the validity of each factor s inclusion in the model. Use of statistically validated efficiency factors in the model ensure the robustness and increase the dependability of the results. Characteristics of the DEA Framework The existing DEA framework for infrastructure allows us to quantify how efficient maintenance units are, which helps in preventing excessive and unrealistic maintenance budget expectations. Following this framework also enables maintenance administrators to include effect of environmental and operational factors on the performance of the road maintenance process (). This framework was followed up by a journal article focusing on the framework s implementation on measuring efficiency of bridge maintenance using DEA (), providing an example for maintenance administrators on how to use the information to assess and compare the performances of a collection of bridge maintenance units. Because of the nature of DEA as a method with no statistical underpinning, there has been skepticism of its implementation in engineering-related field (). Within the engineering field, the linkage of performance measurement to process performance improvement is of great concern. In practice, performance improvement requires that the engineer identify the causal factors that are associated by efficiency performance. However, it was argued that the measurement of efficiency performance is by itself an appropriate and necessary research endeavor. This includes discovering and recognizing patterns of efficiency performance without hypothesizing about the causal factors. Still, engineers have not historically been extensively implementing efficiency measurement concepts to improve design performance of products and transforming performance of production process. Thus, the challenge remains to explore more methodological interface of DEA in engineering, to continue to pursue innovative engineering

5 Juni and Adams DEA applications, especially the ones that offer decision-maker an opportunity to learn about system behavior, and define more effective policies (). RESEARCH METHOD While the limitations of the basic DEA model fits its primary purposes, certain enhancement and modification to the basic DEA modeling process are required to increase the legitimacy of applying DEA method for engineering purposes. This applies true to this study, as it focuses more on identifying characteristics of the efficiency factors as opposed to the efficiency scores themselves. These efficiency factors are the factors contributing to productivity differences between the DMUs. Two-Stage DEA Implementation To improve the existing DEA framework for management of transportation infrastructure maintenance, an enhancement to the original DEA framework called the two-stage DEA (DEA) is proposed. The formal statistical foundation for DEA was provided by Banker in, by identifying conditions under which DEA estimators are statistically consistent and maximizes likelihood (). The typical DEA process started by evaluating productivity of each DMU in the first stage based on data on input consumption and output production, just like performing a basic DEA modeling. The efficiency score from the first stage is then regressed on potential contextual factors in the second stage, to identify the factors whose impact on productivity is statistically significant. After testing several different methods, results show that DEA-based methods provide consistent estimators of impact of contextual variables affecting productivity. It was concluded that the method using DEA in the first stage followed by OLS regression model in the second stage performs best in evaluating the impact of efficiency factors on productivity/efficiency (). Most importantly, DEA provides a formal statistical basis for DEA, as it was able to accurately identify the impact of the various efficiency factors in a statistically consistent fashion. From previous discussions about basic DEA implementation, it was established that in situations where maintenance administrators have little information on the functional form linking the outputs and inputs, DEA is particularly useful to estimate the production correspondence, organizational inefficiency, and factors driving productivity/efficiency. Implementation of DEA method establishes DEA as a stochastic frontier estimation (SFE) method that does not impose a strong parametric structure, unlike other SFE methods (). Implementation of DEA Method on Individual DMUs over Multiple Years Following the approach introduced in 0 (), the implementation of DEA in this study is decentralized, focusing on individual DMUs by comparing individual DMUs efficiency scores from different years to determine which year each DMU performs the best and the worst. The purpose for this is to identify the factors that are most affecting efficiency scores in each individual DMU by searching for potential patterns of these factors within the individual DMUs. This can help maintenance administrators target specific efficiency factors to improve maintenance efficiency in a particular DMU. Implementing DEA this way also ensures that the study observes the different combinations of efficiency factors of an individual DMU throughout several maintenance cycles, observe the efficiency scores of an individual DMUs year after year and how the efficiency factors fluctuate during those years. The goal is to recognize and identify

6 Juni and Adams the efficiency factors in each DMU that appear to have significant effects, either positive or negative, on the DMU s efficiency scores. The difference in the typical use of DEA modeling and how it is implemented in this study is illustrated in Figure (). Diagram on the left shows the typical use of DEA modeling in comparing three different DMUs (A, B, C) with their own values of input and output parameters, in the year 0. The diagram on the right shows DEA modeling as it is performed in this study, comparing DMU A and its set of input and output parameters, in three different years (0, 0, 0). 0 0 FIGURE Typical use of DEA vs. this study (). In summary, the purpose of employing decentralized analysis in this study is because unlike typical DEA implementation, the goal is not simply to identify the most efficient and inefficient DMUs, but to observe individual DMUs and identify and quantify the factors affecting its efficiencies. Identifying these factors will provide maintenance administrators with important information they need to evaluate their past maintenance performance and improve their maintenance strategy for the future. After the factors significantly affecting the efficiency of maintenance work are identified, the results show the efficiency score of each DMU in comparison with other DMUs in a particular model, which in this study means the same DMU, but at different years. A particular year where an observed DMU having an efficiency score of 0% implies that the DMU is doing the best this DMU can do, compared to the other observed years in the model, as a result of the efficiency factors included. On the other hand, a low efficiency score for a particular year simply means that the DMU could do better. ANALYSIS The purpose of the analysis is to introduce the application of two-stage DEA analysis in an implementation of the DEA framework to assess performance of maintenance units. This allows

7 Juni and Adams 0 0 statistical validation of efficiency factors, assisting maintenance administrators in selecting the most appropriate parameters to use in the DEA model. Data and preparation The data used to test this DEA implementation are bridge maintenance data at the region level in Wisconsin from 00 to 0, which are gathered from multiple sources, including Wisconsin DOT and Federal Highway Administration s National Bridge Inventory. With the decentralized analysis looking to identify individual DMUs in a multiple-year period, all the DMUs in the model are the same individual maintenance unit at the region level, with the year marker of each data set being used to identify each DMU, labeled as Year. As the most important input, maintenance expenditures is labeled as Exp, and is shown in dollars spent for bridge maintenance in that year. With the analysis focusing on implementing DEA method on bridges, the other data being used is Sufficiency Rating (SR). To use SR data in this DEA implementation, it needs to go through an adjustment. First, as the SR data is recorded as a bridge-level value while the analysis will be done at the Region level, the SR data needs to be rolled up to the region level. Because of the different sizes of bridges spread out in a particular Region, simply averaging the SR values of all the bridges in the Region does not result in a value that is intuitively representative of all the bridges in the Region. Therefore, rolling up the SR values to the Region level are done by weighting SR in each bridge with the bridge s total deck area. With SR values in individual DMUs changing slightly from year to year as a result of maintenance process and normal deterioration, the changes of SR from the previous year is used as the parameter for DEA, as follows: SR = (SRy- SRy), where y is the current year. A final adjustment for SR data is needed as some of the changes in SR between years are shown to be a negative number, showing how sufficiency rating decreases. Non-positive value is not allowed as a DEA output, however DEA is known to have output translation invariant, so a linear transformation of output data can be performed with no impact in DEA efficiency results, as long as input oriented model is used (). Therefore, the solution for this negative SR values is to add a value to all the SR values in the model that are larger than the smallest negative value in the data set in the model, as shown in Table as an example from NC region. After this adjustment, the SR data is shown on the table as SRadj. TABLE Adding Adjustment Values to SR Year SR SR adj The rest of the efficiency factors included in the model are Winter Severity Index, Average Daily Traffic, and total bridge deck area. Winter Severity Index data is labeled as WSI, Average Daily Traffic is labeled ADT, and Total bridge deck area in square feet is labeled Area. The important thing to notice about these three parameters is that when their

8 Juni and Adams 0 0 values increase, then there is likely to be a decrease of the output parameter SR. This goes against the isotonicity principle of DEA that says a DEA model s input variables should be defined as such that an increase in the input variables is accompanied by an increase in the output variables (). To resolve this issue, the multiplicative inverse of these values are calculated. Multiplicative inverse or reciprocal is denoted by /x, where x is the original value. Using multiplicative inverse values ensure that a direct proportion between the input parameters and the output parameter is established without going against the isotonicity principle. This particular approach is commonly employed in DEA studies (, 0). This adjustment was applied to the data set. After this adjustment, WSI, ADT, and Area are shown on the table as WSIadj, ADTadj, and Areaadj. The final list of parameters ready to be used in the DEA model are: Year, SRadj, Exp, WSIadj, Areaadj. DEA Stage One: Classic DEA modeling In the first stage, the data set was run through a DEA modeling program, resulted in a set of efficiency scores for all of the DMUs, which are all five regions in Wisconsin from 00 to 0, as shown in Table. The calculation includes running the bridge maintenance data on each of the five regions in Wisconsin, one model for each region, for a total of five region-level models. The results from these models are then used to show the important data points in the group of DMUs being observed, which includes information on the year(s) a particular DMU shown as performing efficiently or inefficiently. The parameter data of bridge maintenance in 00 to 0 from the five regions (North Central, North East, North West, Southeast, and Southwest) are shown in Table. After running these parameters as input data for the DEA solver program, the years where the DMUs performed inefficiently were identified. The highlighted rows in Table show the years where a particular DMU received less than 0% efficiency scores, with the efficiency scores shown in the right most column. TABLE Summary of Input/Output Parameters and Efficiency Scores from All Five Regions DMU Efficiency Year SRadj Exp WSIadj ADTadj Areaadj (Region) Score Model (NC) Model (NE) $,, % $,, % 0. $,, % 0 0. $,, % 0 0. $,, % 0.0 $,0, % $,, % $,00, % 0. $,, % $,0, % 0 0. $,0, % 0 0. $,, %

9 Juni and Adams 0 DMU (Region) Model (NW) Model (SE) Model (SW) Year SRadj Exp WSIadj ADTadj Areaadj Efficiency Score 00. $,0, % $,0, % 0. $,, % $,, % 0 0. $,, % 0 0. $,00, % $,, % $,, % 0. $,, % 0 0. $,, % $,0, % $,0, % $,, % $,, % 0. $,0, % $,, % 0 0. $,, % 0.0 $,, % As shown in Table, the model for the North Central (NC) region identifies the year when bridge maintenance were performed inefficiently, which is 00 (.% efficiency). In the North East (NE) region, the model identifies no inefficient year. In the Northwest (NW) region, the model identifies two inefficient years, which are 00 with.% efficiency and 00 with.% efficiency. In the Southeast (SE) region, the model also identifies two inefficient years, which are 0 with % efficiency and 0 with.% efficiency. The Southwest (SW) region has one inefficient year, in 00 (.%). Initial results show that overall the DMUs have been performing good. The NE region has been perfectly efficient during the six-year period, relative to its own performances in all the observed years. While this is a good thing, this also means that the model does not have the ability to discriminate between NE region s performances during the years. This could be a result of one of the issues inherent in DEA modeling, where DEA has difficulties discriminating among the DMUs when the number of DMUs in the model is not sufficiently larger than the total number of variables (). This also means that the method cannot see any difference between the maintenance operations in the NE region during the period, and therefore does not have a way to identify any specific characteristics of the efficiency factors. DEA Stage Two: OLS Regression In the second stage, the results of the first stage (shown in Table ) was regressed on the efficiency factors using ordinary least squares (OLS). For the purpose of this second stage, the factors affecting efficiency are WSIadj, ADTadj, and Areaadj.

10 Juni and Adams 0 0 Results from regressing efficiency scores on the three efficiency factors show the expected inconclusive result from NE region, as all the years show 0% efficiency after the first stage. Analysis of variance (ANOVA) from the regression shows P-values of less than 0.0 in NC and NE regions for ADTadj and Areaadj, but higher than 0.0 in SE and SW regions. Also, all regions are showing a high P-values in for WSIadj. Additionally, F-test of the overall significance for all the regions are showing similar results, with low values for NC and NW, and high values for SE and SW regions. Adjusted R-squared values also support the P-values and F-test result. The list of P-values and overall significance from the regression is shown in Table, with the high values highlighted. TABLE P-values, Overall Significance, and Adjusted R-squared for Regional models Intercept WSIadj ADTadj Areaadj Significance Adjusted R- NC NE N/A N/A N/A N/A N/A N/A NW SE SW Based on table, the set of efficiency factors previously selected are shown as poor in predicting the efficiency scores. In particular, WSIadj is showing that it is unlikely to be a meaningful factor in the model at all, as can be seen in the P-values associated with it. This strongly suggests for WSIadj to be removed from the model in the next step. The models for SE and SW region confirm how the model is unfit, by showing P-values higher than 0.0 for all factors, higher than 0.0 in overall significance, and some very low adjusted R-squared values. Rerun DEA Modeling New DEA models for all the regions are then recalculated with only ADTadj and Areaadj as the efficiency factors. In essence, the removal of WSIadj means foregoing consideration of different weather effect on any of the counties. The explanation for this is that weather pattern might be a good inclusion as an efficiency factor when comparing peer regional areas as DMUs, but as this study focuses on individual DMUs, the weather pattern change between the years in individual DMUs shows that they are not varied enough to affect efficiency of the DMU. The removal of WSIadj also helped the DEA model in decreasing the total number of variables, which was established before as a way to increase DEA model reliability. After removing WSIadj, the DEA models for all the regions are rerun, with the results being shown in Table. TABLE Summary of Input/Output Parameters and Efficiency Scores without WSI Model Efficiency Year SRadj Exp ADTadj Areaadj (Region) Score Model (NC) $,, % $,, % 0. $,, % 0 0. $,, % 0 0. $,, %

11 Juni and Adams Model (Region) Model (NE) Model (NW) Model (SE) Model (SW) Year SRadj Exp ADTadj Areaadj Efficiency Score 0.0 $,0, % $,, % $,00, % 0. $,, % $,0, % 0 0. $,0, % 0 0. $,, % 00. $,0, % $,0, % 0. $,, % $,, % 0 0. $,, % 0 0. $,00, % $,, % $,, % 0. $,, % 0 0. $,, % $,0, % $,0, % $,, % $,, % 0. $,0, % $,, % 0 0. $,, % 0.0 $,, % Table shows the final results of the DEA model. After going through both stages of DEA, the result shows that all regions have at least two times during the six year period where they did not perform efficiently compared to the other years. The lowest efficiency score from all the regions is.%, which is the score of NC region in 00. All the regions were inefficient in 00 or 00. For NE, NW, and SW regions, both 00 and 00 are inefficient years. All the regions are efficient in 0, the last year in the observed period. SUMMARY OF FINDINGS The objective of this study is to propose an improvement to an existing DEA framework for maintenance of transportation infrastructure assets by introducing the use of two-stage DEA method. This improvement involves regressing the values of the efficiency factors and testing their statistical viability and rerunning them through the DEA model.

12 Juni and Adams 0 0 DEA results The inclusion of statistical aspects with the DEA method helped in identifying efficiency factors that are significantly affecting efficiencies of maintenance unit. Based on the result, an efficiency factor that was previously used (WSIadj) was removed from the model, as it was statistically proven to be insignificant as an estimator of efficiency. This removal of an insignificant efficiency factor increases the validity of the efficiency numbers resulted from the efficiency comparison after the model was rerun. Table shows the comparison of the efficiency scores calculated by the DEA models, before and after the removal of WSIadj as an efficiency factor. The difference in efficiency scores ranges between 0.% and.%. While most of the differences are relatively low in percentage, what is most important here is the fact that there are DMUs that were previously identified as efficient (0%), turned out to be inefficient (non-0%) after WSIadj is removed. This is a significant change in terms of DEA modeling, as to further use the results of the model to identify the characteristics of these efficiency factors, the focus will be on DMUs that achieve 0% efficiency. With the exclusion of an efficiency factor that is proven to be insignificant, a potential misperception of the effects of these efficiency factors was avoided. TABLE Comparison of Efficiency Scores Before and After Removal of WSIadj Model Efficiency Score Efficiency Score Efficiency Score Year (Region) (with WSIadj) (without WSIadj) Differences Model (NC) 00.%.%.% 0 0%.%.% Model (NE) 00 0%.% 0.% 00 0%.%.% 00.% % 0.% Model (NW) 00.%.% 0.% 00 0%.% 0.% 0 %.% 0.% 00 0%.%.% Model (SW) 00.%.% 0.% 0 0%.%.% Efficiency Factors Selection For the purpose of implementing DEA method in transportation infrastructure maintenance, the fact that doing a two-stage analysis helps in filtering out efficiency factors, confirming the significant ones and exposing the insignificant ones is critical. Previously, research efforts implementing DEA method did not use any statistical validation. Selection of efficiency factors has been rationalized by logical explanation and based on experience in the real world. Having a method that will confirm these rationalizations helps a lot in validating the use of this method in engineering-related field.

13 Juni and Adams 0 0 FOLLOW UP Improving the Model There are a few improvements to further improve the implementation of DEA in transportation infrastructure maintenance:. The DEA modeling implemented above was able to come up with efficiency scores for all DMUs and identifies WSIadj as an insignificant efficiency factor for the model. To further increase the validity of the model, the next step that could be done is to implement a DEA bootstrapping technique to the original data set. Several examples of bootstrapping technique for DEA is available, most notably from Simar and Wilson (). Bootstrapping increases the size of the data set, allowing for more accurate estimate of bias and variance, and to construct confidence interval (). This is especially helpful when combined with the two-stage DEA method where the efficiency scores calculated in the first stage is regressed on the efficiency factors in the second stage. Naturally, more sample size in both stages will lead to more discriminating power of the model, leading to better representation of the DMUs, and therefore more accurate calculation of efficiencies of the DMUs.. Investigate the possibilities of including target values as an efficiency factor in the observed DMUs. Target is established by maintenance administrators as the condition level to achieve, and the decision to establish a target level may have a technical, economic, or policy reasons. The idea that needs to be tested is that maintenance operations in the field relies heavily on the determined target values, to the point where some units may work more efficiently when they know that a particular target is set high, or less efficiently when they know that a particular target is set low. More about this is discussed in the Parameters Selection section. Sensitivity Analysis The next goal is to identify and quantify the characteristics of the efficiency factors. There are a few examples of DEA-based sensitivity analysis (). Doing a sensitivity analysis will improve the stability of classification of DMUs and help correctly identify efficient and inefficient maintenance units.

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