A Data Envelopment Analysis approach for accessibility measures: Simulating operational enhancement scenarios for railway across Europe

This article summarizes the results of the model simulations carried out in order to estimate the potential impacts on rail accessibility of infrastructure enhancements across Europe (see also [17]). The model simulations have been performed by a combination of the TRANS-TOOLS rail network and Traffic Analyst: the TRANS-TOOLS (“Tools for transport forecasting and scenario testing”) is a European transport network model that has been developed in collaborative projects funded by the European Commission’s Joint Research Centre and DG TREN (for more information see [49]) and it provides a quite detailed European rail network for 2005; its assignment module (Traffic Analyst by Rapidis, see [50]) allows the model to capture changes in route choice as a result of hypothesized changes in speed and frequency.

The data for origins and destinations at NUTS 3 level (the Nomenclature of Units for Territorial Statistics is a geocode standard developed and regulated by the European Union) and for the baseline (2005) have been assumed according to the ETIS Plus figures (downloadable at [51]): ETIS Plus is a FP7 project on data collection for transport at European level aiming at providing a bridge between official statistics and applications within the transport policy theme; in practice it consists of an European Transport Policy Information System, combining data, analytical modelling with maps (GIS), and a single online interface for accessing the data.

The post-processing analyses of the results have been carried out with utilities developed in Matlab, while the outcomes for each zone have been also reported in easy-to-read ArcGIS maps (Fig. 1).

Beyond the baseline (2005), three different scenarios have been implemented by changing speeds as follows:

In practice the article assumes the best and worst (hypothetic) network settings by simulating respectively the Scenario 200 km/h and Scenario 45 km/h; this last one hypothesizes a degradation of the current network (or better an imaginary configuration previous to the baseline), to evaluate the benefits of the current infrastructure endowment compared to this lower bound. Subsequently the analysis estimates the effects of a more feasible and realistic interventions such as increases of speed of some links (with current speed lower than 90 km/h).

As also proposed in other studies [18] our analysis considers the centroids of NUTS3 regions as origins and destinations. The all-or-nothing assignment module calculates the minimum paths through the networks, i.e. the path with minimum travel times between the centroids of the regions. According to the new travel times between each couple OD, for each scenario various accessibility indexes have been evaluated.

In the last decades, in fact, accessibility has been measured by several types of indicators often based on two different concepts: travel resistance (cost, time, etc.) and attractiveness of urban agglomerations (depending on variables such as population, employment or gross domestic product). In this analysis we have considered four different accessibility measures offering complementary information and highlighting different cost or attraction attributes.

The location index represents the average travel time between each couple OD weighted on the mass, measured in this analysis by population of the destination regions:where:

Since no distance decay function (and so no discrimination between neighbor or far locations) is considered, the accessibility for each zone depends on the geographical position; remote locations present low accessibility values and even a good transport infrastructure endowment could be not enough to overcome the negative effects of a large geographical distance to the main activity areas [11]. Figure 2 reports the location index for each NUT3 zone in Europe and for each simulated scenario showing clearly the above described core-periphery patterns; as expected the scenarios with maximum speed of 45 km/h or minimum speed of 200 km/h on the whole railway network present respectively the highest and lowest values of the location index.

The network efficiency indicator “represents the distance between the real accessibility against the best accessibility that can be obtained if the zone is connected with all the other regions by the best possible infrastructure” [22], in our case a network with speed on each link of at least 200 km/h.

It offers a measure in terms of the relative ease of access according to the network efficiency; the relative ease of access is represented by the ratio of the travel time between 2 zones to the ideal travel time between the same zones assuming the best possible infrastructure, i.e. for us scenario 200 km/h):where:

This indicator provides an idea on how efficient are the connections from a given zone, regardless of its geographic location: it could occur that a region which is peripheral according to the location index is highly accessible in terms of network efficiency [11]. The following figure shows the values of this indicator for each zone and each scenario; of course the scenario with speed of at least 200 km/h on the whole network represents the best possible (ideal) setting (A = 1 for each region) (Fig. 3).

Finally regarding the potential and the daily accessibility, it is possible to express them as a construct of two functions: the activities function (representing the activities or opportunities to be reached) and the impedance function (representing the effort, time, distance or cost needed to reach them):where:

In practice (3) calculates the total of activities reachable in j weighted by the ease of getting from i to j. As described by the impedance function, the interaction between locations declines with the increasing disutility (distance, time, and/or costs) between them. In general, the perception and valuation of the distance between an origin and a destination differ according to transport modes, purpose of trips, characteristics of the household and of the destination [3]; in the present paper we focused on rail mode and on the population of destination.

Several forms of distance decay function have been already used and described in past accessibility studies; this analysis considers two different shapes depending on travel time (Fig. 4):

Of course different shapes of the impedance function could represent diverse aspects (and could provide different indications) of accessibility; the described exponential decay function (potential accessibility) allows to consider the population (activities) of all the reachable zones even if with a diverse weight depending on travel time, while the proposed ‘ad hoc’ decay function associates the accessibility measures only to short trips with travel time within 4 h (allowing ‘daily’ commuting) giving a different and more specific indication. The following figures report the potential and daily accessibility values for each NUT3 zone and each considered scenario.

To allow a first analysis of the results, Table 1 reports the percentage of variation (at country level) of all the considered accessibility indicators comparing each alternative scenario and the baseline 2005 (do-nothing configuration); it presents also the average speed on the network (for each country) weighted on the length of the links.

Analyzing the results of the previous table and the Figs. 2, 3, 5 and 6 it is not difficult to recognize the different approach of each indicator, such as for example the core-periphery and border patterns of the location index and of the potential accessibility.

As pointed out in [24], in fact, “the interpretation of the partial accessibility indicators is clear and useful for policy-makers, but it does not provide a synthetic and global measure that allows planners to compare or fully rank the level of accessibility for different regions or cities within Europe. Besides this, it is necessary to control for some contradictory results that may confuse planners who only use a partial approach.”

Table 2 reports the Pearson and the Spearman correlation coefficients among of the considered accessibility indexes and for each scenario to assess how well their relationship can be described using respectively a linear or a monotonic function; results seem to justify the assumption that the different ‘partial’ indicators can be considered complementary more than substitutive. As expected the potential and the daily accessibility indexes present the strongest correlation; we have retained both the indicators in our successive analyses since, how already noticed, they allow us to measure accessibility from different perspectives (overall or ‘within-4-h/daily’ activities reachable from each zone).

To better analyze the accessibility impacts of the simulated operational enhancements and trying to summarize the complementary information provided by the four described indicators, the authors have attempted to explore the construction of a composite indicator (CI) by means of two different approaches:

The first methodology is based mainly on the estimation of a DEA-accessibility index as already proposed by Martin, Gutiérrez, Roman and also Reggiani [22‐25] while the second method recalls the wide scientific literature for creating composite indicators in analysis at large scale [26‐36].

A frequent issue concerning the development of a global index is related to the weights to assign to the partial indicators; in this context, the main appeal of DEA-based composite indicators (CIs) is that they “look for endogenous weights, which maximize the overall score for each decision-making unit given a set of other observations” [31].

Indeed the Data Envelopment Analysis is a non-parametric methodology for evaluating the relative efficiency of Decision Making Units; practically the proposed DEA approach suggests solving the following multiple objective problem of accessibility for both the scenarios with and without interventions:where j represents the generic European NUT3 zone and L, N, PA and PDA represent respectively the location index, the network efficiency indicator, the potential and the daily accessibility.

In other words this “synthetic DEA-based indicator is calculated as the inverse of the maximum proportional accessibility outputs that can be obtained for the indicated accessibility inputs” [10].

The model determines for each scenario the most efficient zones (from an accessibility perspective) to individuate the frontier of the envelopment surface; the regions not lying on the frontier are inefficient and the measurement of the grade of inefficiency is represented by their distance from this ‘best-practice’ frontier.

When the data cannot be easily interpreted as inputs or outputs, a general rule suggests to consider the variables for which lower levels are better as inputs (in our case Location index and Network efficiency), and to treat as outputs those variables for which higher amounts are better (potential and daily accessibility in our analysis) [10].

We have assumed variable returns to scale (VRS) because of the great heterogeneity among of the various EU regions and an output orientation.

The Benefit of Doubts approach, instead, is rooted in the copious literature concerning CIs. Considering m sub-indexes and n regions the idea is to merge the sub-indicators’ values per region into a single number, e.g. their weighted average. In absence of reliable information about the weights, the proposed approach endogenously determines the weights maximizing the CI value for each region. In practice it comes to solving the following linear programming problem for each zone j:where:

The benefit-of-the-doubt interpretation of the methodology resides in the fact that the method chooses those weights maximizing the resulting indicator: the highest relative weights are accorded to those sub-indicators for which the zone j achieves the best performance (in relative terms) when compared to the other regions in the sample.

This model is equivalent to the original input oriented, constant-returns-to-scale DEA model presented by Charnes et al. [33] (for more details see [29]), with the sub-indicators representing the different outputs and a single ‘dummy input’ with value unity to each country.

Although, as pointed out in [30], a BOD-based indicator meets the property of units invariance, which makes the normalization stage redundant, the selected four sub-indicators have not the same direction, meaning that for some of them (location index and network efficiency) higher values represent worse performances. To overcome this issue the “min-max” normalization method has been applied to the results of the partial indexes according to the original direction of the variables: where:

To better compare and also to test the two proposed approaches, sensitivity and robustness analyses are performed on both the aggregate indicators for the baseline 2005 (do-nothing scenario). We have carried out the Data Envelopment Analysis several times eliminating the sub-indicators one by one, to evaluate the weight, the importance associated to each dimension in both the procedures and also to estimate the changes in scores.

Tables 3, 4, 5 and 6 report the Spearman correlation coefficients across the sub-indicators (eliminating one of them per time) and across the scenarios respectively applying the DEA or the BOD approach. In particular:

The tables point out the different set of “endogenous” weights (and the consequent relevance) associated to the sub-indicators applying the two methods and also how they vary across the scenarios; while for the DEA the most relevant indicator comes to be the potential accessibility, for the BOD procedure the major changes in results are obtained eliminating the location index. This is due to the different approach of the methods; as noticed by Martin and Reggiani [24], the DEA results “can be interpreted as the distance that separates every region from the most accessible location that can be found taking into account all the observations”. In other words this “indicator measures how accessible a region is with respect to all the other regions included in the sample”. The BOD approach, instead, maximizes the CI value (see formula (6)) for each region and so it takes in account also the heterogeneity of the sub-indicators among of the regions: the highest relative weights are accorded to those dimensions for which the region j achieves the best performance (in relative terms) when compared to the other regions in the sample; in other words “each region has its own weights which are optimal and guarantee the best possible position for the associated region among all other regions in the sample. With any other weighting scheme, the relative position of that region would be worse.”[35].

The main disadvantage of this last method is that without additional setting constraints countries performing very well only in one indicator can be considered successful. This is why the scientific literature has focused on different ways to take in account experts’ (subjective) opinions or to set additional relative weight constrictions (e.g. pie share constraints). Anyway we have explored the original BOD approach and so no additional constraints are added except those already described in (6). The described effect is quite evident in Tables 5 and 6 where the Scenario 200 is less relevant than the others since the network efficiency index presents unity value for each region (reference “best-possible” infrastructure).

Finally a robustness analysis has been performed on both the synthetic indicators adding a uniform casual noise to the normalized value of each sub-indicator, according to the formula:where

In practice we have performed a Monte Carlo simulation by extracting 100 random values (within the range defined) of noise for each sub-indicator and assessing the related variation in outcomes. To evaluate the dispersion of the results, for each region we have calculated the variation coefficient, defined as the ratio of the standard deviation to the mean:where

Table 7 reports the mean and the standard deviation across all regions of the variation coefficient C _Vj . Both the approaches seem to be robust against the assumed uniform casual noise: by hypothesizing a value of α equal to 5 % the results show a variation coefficient of less than 2 % in both the cases.