improving the efficiency of metropolitan area transit by joint analysis of its multiple providers

Transportation Research Part E 47 (2011) 1160–1176

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Transportation Research Part E

journal homepage: www.elsevier .com/locate / t re

Improving the efficiency of metropolitan area transit by joint analysisof its multiple providers

Darold T. Barnum a,⇑, Matthew G. Karlaftis b, Sonali Tandon c

a Departments of Managerial Studies, Information & Decision Sciences, and Pharmacy Administration, University of Illinois at Chicago (MC 243),601 South Morgan Street, Chicago, IL 60607-7123, USAb Department of Transportation Planning Engineering, School of Civil Engineering, National Technical University of Athens, 5, Iroon Polytechniou Street,Zografou Campus, Athens 15773, Greecec Chicago Transit Authority, 567 W. Lake Street, Chicago, IL 60661, USA

a r t i c l e i n f o

Article history:Received 24 August 2009Received in revised form 16 May 2010Accepted 18 July 2010

Keywords:Data Envelopment AnalysisDEATechnical efficiencyAllocation efficiencyUrban transitUrban public transportationUrban transport

1366-5545/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.tre.2011.04.006

⇑ Corresponding author. Tel.: +1 312 996 3073; faE-mail addresses: [email protected] (D.T. Barnum

a b s t r a c t

Public transportation in a metropolitan area often is supplied by multiple types of transit.This paper develops and illustrates a DEA-based procedure for estimating: overall effi-ciency of an area’s public transportation; technical efficiencies of the individual transittypes; effect of each type on overall efficiency; and efficiency of the allocation of resourcesamong types and an algorithm for improving it. The paper concludes that the overall effi-ciency of an urban area’s public transportation can be validly estimated only if the techni-cal efficiency of each major transport type and the efficiency in allocating resources amongthem are taken into consideration.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Efficiency has long been a critical consideration in both policy and operational decisions of urban transit systems, andtransit efficiency has recently become even more important. In the US, for example, transit ridership has been increasingwhile tax-supported funds to cover its expenses have been declining (Fausset, 2009; Murray, 2010; Torbati, 2010). Of course,fares can be increased and service decreased in response to these lost resources. But, such actions especially hurt those whoare unemployed and those who are transit-dependent. And, cuts in service and ridership negatively impact efforts to de-crease the use of oil and reverse undesirable climate trends. The effects of declining revenues can be lessened if efficiencyis improved. Indeed, the desirable effects on energy and climate would be multiplied many times over if cities across theworld could improve the efficiency of their transit systems.

Because of efficiency’s long-standing importance to government, there have been many US federal, state and local studiesthat compare the efficiency of transit systems, with a few recent publications including (Perk and Kamp, 2004; Stanley andHendren, 2004). Further, more than 60 transit Data Envelopment Analysis (DEA) papers have been published or are in pressas of mid-2009, involving cities in Asia, Europe, and North and South America. Perhaps as a sign of the growing importance oftransit efficiency, over half of these DEA studies have come in the last six years (Barnum, 2009), including six in Transpor-tation Research Part E (Boame, 2004; Sheth et al., 2007; Graham, 2008; Lin et al., 2008; Yu and Fan, 2009; von Hirschhausenand Cullmann, 2010).

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x: +1 312 996 3559.), [email protected] (M.G. Karlaftis), [email protected] (S. Tandon).

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D.T. Barnum et al. / Transportation Research Part E 47 (2011) 1160–1176 1161

2. Joint analysis of multiple transportation modes

Multiple types of public transportation, overseen by metropolitan transit agencies, serve most urban areas. Therefore, tobest improve the efficiency of an urban area’s transit service, it is necessary to individually and jointly analyze the variousmethods by which that service is supplied. That is, it is necessary to analyze the individual efficiencies of the main methodsof delivering public transportation in urban areas, and synthesize the individual results to show their effects on the efficiencyof an urban area’s public transportation as an integrated whole. Specifically, in order to provide government policy makersand transit managements with sufficient information, it is necessary to:

� Estimate technical efficiency of each type of transit service provided by a transit agency.� Estimate efficiency of the agency as a whole in supplying service to its urban area.� Estimate effect of changes in each of service type’s efficiency on its parent agency’s efficiency.� Estimate efficiency of the parent agency’s allocation of resources among service types.� Estimate reallocation of resources among the service types that minimizes the parent agency’s total costs while maintain-

ing its total output.

Without all of this information, decision makers cannot comprehensively evaluate and improve the transit service in ametropolitan area.

3. Review of the literature

Unfortunately, 56 of the 63 transit DEA articles published through mid-2009 deal only with one mode, almost alwaysmotorbuses (Barnum, 2009). The remaining publications are discussed next.

The earliest transit DEA articles to consider multiple modes were published in 1997 and 1998, involving fixed-route,fixed-schedule motorbuses and demand-responsive (paratransit) operations (Viton, 1997; Viton, 1998). Outputs and mostinputs were entered as separate variables for each of the modes. So, for example, an agency’s four outputs were bus vehi-cle-miles, paratransit vehicle-miles, bus passenger-trips, and paratransit passenger-trips. For the most part, the inputswere also separated by mode. One DEA score was reported for each agency, and the two modes were not analyzedseparately.

The next set of transit DEA articles that considered multiple modes were published in 2006 and 2008, and involved orga-nizations that operated both highway and urban bus lines (Yu and Fan, 2006; Yu, 2008). Again, the values of most input andoutput variables were entered separately for each mode. There were also a shared input and a shared output in the earlierpaper, and a shared input in the later. The values of the shared variables were artificially allocated to the two modes in a waythat maximized the efficiency score for the organization as a whole. There were analyses of the organizations as a wholeusing all inputs and all outputs, and analyses of each mode separately using only variables attributed to that mode (includingthe amount of the shared variable that had been previously artificially allocated to it).

In Yu and Fan’s, 2009 paper in Transportation Research Part E, again the non-shared input and output values were enteredseparately for highway and urban bus lines, and again shared inputs were artificially allocated among the modes in order tomaximize the organizations’ overall efficiency scores (Yu and Fan, 2009). In addition, they presented a network model inwhich the outputs from the first stage were used as the inputs into the second stage. They presented a DEA score for eachorganization, as well as DEA scores for each mode at each stage, using input and output variables applicable to each modeand stage including the artificially allocated amounts.

In 2008 Sampaio, Neto and Sampaio published a paper using DEA to analyze the aggregated inputs and outputs from themajor public transportation modes in large urban areas from around the world. Because they used data aggregated acrossmultiple modes, they identified the overall efficiency of public transportation in each urban area, but not the efficiency ofindividual components (Sampaio et al., 2008).

Barnum, Gleason, Hemily and others published a set of papers involving demand-responsive transit, using separate inputand output values for publicly owned service and for service outsourced to private operators (Barnum et al., 2008, 2009,2010b). These papers did not address the multiple mode issue, however, with their topic of concern being the use of statis-tical panel data analysis to estimate valid confidence intervals and trends of individual DMU efficiencies.

As a whole, these papers leave a number of issues unresolved, and, in some cases, contain errors that invalidate theirmethodologies. All but one of the papers compute DEA scores measuring technical efficiency only, because they utilize sep-arate input and output values for each mode (Barnum and Gleason, 2006a,b). Only Sampaio, Neto and Sampaio’s DEA aggre-gates the inputs and outputs across modes, and therefore is the only one to measure overall total efficiency because itencompasses both allocation and technical efficiency in its DEA scores. Although technical efficiency is important, it doesnot tell the whole story because allocation efficiency is ignored.

Further, with the exception of the papers by Yu and his colleagues, these past papers only report each organization’s tech-nical efficiency, thereby not identifying each separate subunit’s contribution. Clearly, in attempts to improve an organiza-tion’s overall technical efficiency as well as the efficiency of each subunit, it is necessary to identify the technicalefficiency of each subunit. So, while Yu et al. do not provide a method for reporting total (technical and allocational)

1162 D.T. Barnum et al. / Transportation Research Part E 47 (2011) 1160–1176

organizational efficiency as do Sampaio et al., they do attempt to estimate the technical efficiency of each subunit, namely tocompute the DEA score of each subunit alone when compared only to subunits of the same type in other organizations.

Although Yu et al. do compare each subunit to others of the same type, unfortunately their method involves artificiallyallocating shared resources to each subunit, which is not valid, so their efficiency estimates are erroneous. As was first dem-onstrated by Beasley (2003), it is only the actual allocation of a resource that can affect true efficiency. While a shared re-source can be artificially allocated to maximize the reported efficiency, it does not measure true efficiency resulting from theactual allocation of the shared resource.

A simple example can show this fact. Suppose there are two subunits, each producing six units of output and togetherusing 12 units of input. Assume that, in truth, each subunit uses six units of input. Then, the true average efficiency forthe organization is (6/6 + 6/6)/2 = 1. Now, instead of using the actual allocation of the input, let us artificially allocate it inorder to increase the reported efficiency of the organization. One possibility would be to allocate only three units of inputto the first subunit and the remaining nine to the second. This artificial allocation yields an average organizational efficiencyof (6/3 + 6/9)/2 = 1.33, an improvement of 33%. If input units cannot be subdivided, then the maximum reported efficiencywill occur when one subunit or the other is artificially assigned one input unit, for example (6/11 + 6/1)/2 = 3.5. Not only isthe overall efficiency score invalid, but also the efficiency of each subunit is also incorrect. The same problems occur withDEA estimations.

In short, only Yu et al.’s papers even attempt to identify the individual efficiencies of each subunit employed by a metro-politan transportation agency, and they use an invalid methodology. Only Sampaio, Neto and Sampaio’s article aggregatesthe inputs and outputs across modes, and therefore is the only one to encompass total (allocation and technical) efficiencyin its DEA score, but it does not attempt to analyze the efficiencies of the individual modes making up the overall score. Noneof the papers estimate the effect of each subunit’s efficiency on the overall efficiency for the urban area. None estimate theefficiency of the metropolitan transit agency’s allocation of funds among its subunits, and none offer a method for reallocat-ing resources in order to improve efficiency.

A precursor article developed the basic methodology for estimating all of the components of allocation and technical effi-ciency that are used in this paper (Barnum and Gleason, 2010a). In fact, research for this paper on transit efficiency was al-ready underway when we discovered that new theoretical and methodological work concerning allocation efficiency wasnecessary to build on the less-complex methods that had been developed earlier by Barnum and Gleason (2005; 2006b,a;2007). After the necessary theory had been developed (Barnum and Gleason, 2010a), we resumed work on this paper to illus-trate its detailed application to urban transit. In doing so, we greatly refine, extend and apply the basic methods suggested inthe 2010 paper. Further, new coauthors were added to this paper, one working in the transit industry and the other in trans-portation engineering. Both made substantial, multiple changes so this application paper would more closely mirror real-world conditions in the transit industry.

4. Purpose and organization of this paper

In this paper, we analyze the individual efficiencies of the four most common methods of delivering public transportationin United States urban areas, and synthesize the individual results to show their effects on the efficiency of an urban area’spublic transportation as an integrated whole. We treat each metropolitan transit agency as a Decision Making Unit (DMU),with each DMU supplying service in their cities with up to four major types of subunits: self-operated motorbus, outsourcedmotorbus, self-operated paratransit, and outsourced paratransit.

This paper’s goal is to provide transit management decision makers and governmental policy makers with a comprehen-sive DEA procedure for analyzing and improving the efficiency of transit service in a metropolitan area. Utilization of thisprocedure should result in increases in agency efficiency, thereby lessening the extent to which a city’s transit service mustbe cut and fares raised. This would be of significant value to governments, transit management, riders, and, indeed, all cit-izens, because of transit’s environmental impacts. Further, because many aspects of our methodology are new to the DEAliterature as well as to the transportation literature, as identified at applicable places in the paper, the procedure wouldbe of value in DEA analyses of other organizations with multiple subunits.

5. Methods

The model of a public transportation system’s production subunits is shown in Fig. 1.As discussed later, we use operating expenses as the sole input to each of the subunits and seat-hours as the sole output,

so the input is allocable among subunits that all produce the same output. Kao (2009b) has used the term parallel productionsystems for this type of technology. According to the typology recently suggested by Castelli, Pesenti and Ukovich (Castelliet al., 2010), this is a shared flow model, with their other two categories being multilevel models (DMU inputs and outputsare not limited to the inputs and outputs of its subunits), and network models (at least one output of a subunit is input toa different subunit).

In order to identify the sources of a target agency’s efficiency level, a number of tasks are necessary, involving the agencyas a whole and each of its organizational subunits that supply transit service. In all DEAs, the target agency or target subunitis compared only to its peers. For target agency efficiency, all agencies in the sample are included in the DEA. For target

Fig. 1. Urban transit system of parallel production.

1. Estimate the functional relationships between each output and its inputs, and use estimates to choose appropriate DEA models

2. Estimate efficiency of each subunit of target agency

3. Estimate target agency efficiency

4. Estimate effect of changes in each of service type’s efficiency on its parent agency’s efficiency

5. Estimate target agency's allocation efficiency

6. Estimate resource reallocations that minimize target agency costs while maintaining output

Fig. 2. Algorithm flowchart.


subunit efficiency, only subunits of the same type are included in the comparison. We use an input orientation (inputs areminimized while holding outputs constant). The methodology is based on a multiple-task algorithm (Fig. 2).


Each of the algorithm’s six tasks (shown in the boxes) yields estimates. The first task estimates the functional relation-ships between the inputs and outputs. The second through fifth tasks provide estimates of various DEA efficiency indicators.The sixth task, which involves a non-DEA mathematical program, estimates how inputs should be allocated among the targetagency’s subunits in order to minimize total expenses, while holding output constant. In the second through fifth tasks, eachtarget DMU’s outputs and inputs are those actually reported by the DMU and therefore are constants, as is always the casewith DEA. In the sixth task, which, as already mentioned, does not involve DEA, total agency output (seat-hours of service) isheld constant, but input (total operating expenses) becomes a variable to be minimized by reallocating the funds among thesubunits. The suggested order for performing the six tasks is the one that we feel is the most logical and efficient. But, theestimates produced by a task do not usually become direct inputs into later tasks, nor is there any feedback from later toearlier tasks. The algorithm can be viewed as a comprehensive audit of the target agency’s efficiency at a given point in time,to be repeated on a periodic basis.

Each of the tasks is discussed next. For each task, we identify procedures already in the literature. When extant proce-dures are absent or insufficient, we develop a new procedure.

5.1. Estimate functional relationships

Before embarking on DEA, it is necessary to estimate the relationships between the outputs and inputs of each type ofsubunit, and the relationship between the aggregated outputs and aggregated inputs for the agencies as a whole. (We thinkthat similar empirical estimations should precede every DEA, so this prerequisite is not unique to this paper.) If the output-input relationship is linear with a zero intercept, then a constant-returns-to scale DEA model should be used. If the output isincreasing at a decreasing rate, or is linear with a positive intercept, then a variable-returns-to-scale DEA model should beused. If the output is increasing at an increasing rate, then DEA models accounting for non-convexity must be used (Bankerand Maindiratta, 1986; Petersen, 1990; Dekker and Post, 2001; Post, 2001). We estimate the relationships with statisticalPanel Data Analysis models (Washington et al., 2010).

Estimating functional relationships is done only once, before any DEAs are conducted. However, the remaining tasks de-scribed below all must be repeated for each target agency of interest. So, in reading the following tasks, please keep in mindthat only one target agency and its subunits are involved.

5.2. Estimate efficiency of each subunit of the target agency

For each subunit type in the target agency, compute its subunit efficiency, using each subunit’s own inputs and outputs. So,if the target agency supports four subunit types, then there will be an independent DEA score for each of the four. Scorestherefore identify each subunit’s technical efficiency when compared to the same type of subunits in other agencies.

This procedure is new to the DEA literature for identifying subunit efficiency, although necessary for the present case. Inpast work, all subunits have been included in a single DEA rather than conducting a separate DEA for each subunit type. Insome cases, each subunit’s inputs and outputs have a different set of multiplier weights (Färe et al., 1997), while in others allsubunits within a single DMU have the same set of multiplier weights for their variables (Kao, 2009a). Neither solution isappropriate when the subunits are independent, just as neither alternative would be appropriate to apply to independentDMUs. In both cases, the weights assigned to each subunit are not those optimal for its efficiency score, but optimal for max-imizing its parent DMU’s efficiency score. Therefore, one would not be able to identify the true efficiencies of each subunitbecause weights that are not optimal from that subunit’s point of view have been assigned to it. Therefore one cannot validlyevaluate how the target subunit’s performance compares to other subunits of the same type.

Our DEA Program 1 is input oriented and assumes variable-returns-to-scale. For each observation j = 1, . . ., J there are dataon one input and one output. The DEA score h estimates the technical efficiency (inefficient < 1, efficient = 1, supereffi-cient > 1) of the target subunit, in this case subunit k. The target subunit is the one currently being analyzed by DEA. AllDEA programs analyze one target at a time, so the program must be re-run for each individual subunit to be analyzed.

mink

h ð1:0Þ

subject toXJ

j¼1

xjkj 6 hxk ð1:1Þ

XJ

j¼1

yjkj P yk ð1:2Þ

kk ¼ 0 ð1:3Þkj P 0 j ¼ 1; . . . J; j–k ð1:4ÞXJ

j¼1

kj ¼ 1 ð1:5Þ


It is important to understand that these scores identify each subunit’s technical efficiency relative to the same type of sub-unit in other agencies. The goal of the current task, Task 2, is to compare subunits of the same type, so subunit managementcan be correctly evaluated. But, different types of subunits within one agency cannot be compared to each other with thesescores. Comparisons of subunits within the same agency are produced by Task 6, where the goal is to reallocate a givenagency’s inputs among its subunits in order to improve its efficiency.

5.3. Estimate target agency efficiency

For the target transit agency, compute its agency efficiency (all of the target agency’s subunit inputs and outputs aggre-gated). The resulting score estimates the overall (technical and allocative) efficiency with which transit service is providedto the metropolis served by the agency. Recall that in the present case we are considering four different types of subunits, so,if there are I agencies i = (1, . . ., I) and four types of subunit M possible in each agency m = (1, . . ., M), then the aggregatedinput Xi for agency i is:

Xi ¼X4

m¼1

xim ð2Þ

And aggregated output Yi for agency i is:

Yi ¼X4

m¼1

yim ð3Þ

So the DEA program for the target agency k becomes

mink

h ð4:0Þ

subject toXI

i¼1

Xiki 6 hXk ð4:1Þ

XI

i¼1

Yiki P Yk ð4:2Þ

kk ¼ 0 ð4:3Þki P 0 i ¼ 1; . . . I; i–k ð4:4ÞXI

i¼1

ki ¼ 1 ð4:5Þ

5.4. Estimate effect of changes in each service type’s efficiency on its parent agency’s efficiency

To our knowledge, no one in the past has attempted to specifically identify the impact of each subunit on its DMU’s effi-ciency. The importance of identifying the impact of each subunit on its agency’s efficiency is that even if a given subunit isquite inefficient (superefficient), if it has minimal impact on the agency’s overall efficiency, then it may not be worth thetrouble to try to improve (maintain) it, and vice versa for a large impact.

Because we are interested in the impact of each individual subunit, the procedure must be completed separately for eachsubunit whose efficiency is not exactly 1, that is, each inefficient or superefficient subunit. So, for the target agency k and itstarget subunit m, first compute the maximum level of input that would make the target subunit m exactly efficient (h = 1).Recall that hkm already has been computed in Task 1 above. So, with the first subscript being the target agency and the secondsubscript being its subunit m that is the current target,

x�km ¼ hkmxkm ð5:0Þ

Note that x�km < xkm for an inefficient target subunit m, and x�km > xkm for a superefficient target subunit m. Next, compute thenew aggregated input for the target agency:

Xm�

k ¼ ðx�km � xkmÞ þX4

m¼1

xkm ð5:1Þ

where Xs�k is the aggregated input of the target agency k when the input of the target subunit m has been increased or de-

creased so that its h = 1. Now, recompute the DEA for the target agency k:


mink

u ð5:2Þ

subject toXI

i¼1

Xiki 6 uXs�k ð5:3Þ

XI

i¼1

Yiki P Yk ð5:4Þ


i¼1

ki ¼ 1 ð5:7Þ

The change in target agency k’s efficiency (D) resulting from its subunit m becoming exactly efficient is:

D ¼ u� h ð5:8Þ

With efficiency improving (D positive) when target subunit m was originally inefficient and efficiency declining (D negative)when the target subunit m was originally superefficient. Again, recall that the Linear Program 5 must be reused for each sub-unit whose h – 1.

5.5. Estimate target agency’s allocation efficiency

There have been few attempts to estimate the efficiency with which DMUs allocate resources to their subunits, that is, toidentify a DMU’s relative allocation efficiency when compared to other DMUs in the sample. Nesterenko and Zelenyuk (2007)elegantly develop group efficiency measures that include indicators for measuring the efficiency with which allocable re-sources are allocated within the group. However, their model requires many assumptions that are not met by the currentsituation. More applicable is work by Barnum and Gleason (2005; 2006b,a; 2007, 2010a)) that measures the allocation inef-ficiencies resulting when a DMU with multiple outputs and multiple inputs misallocates the share of each input that is givento each output. The process we use here is adapted from their approach, because our outputs are produced by multiple sub-units, which are equivalent to multiple production processes.

Applying the procedure, in order to measure allocative efficiency between our DMUs, it is necessary to decrease the in-puts of every inefficient subunit from all DMUs in the sample so that all subunits will reside on the production frontier oftheir subunit type, then recompute the target DMU’s efficiency score. Once all subunits of every DMU reside on their subunittype’s production frontier, the only differences that remain in their aggregated efficiency will be those caused by inefficientallocation of inputs among the subunits. Therefore, the efficient input ~xim for agency i’s subunit m is:

~xim ¼ himxim h < 1; i ¼ 1; . . . ; I; m ¼ 1; . . . M ð6:0Þ~xim ¼ xim h P 1; i ¼ 1; . . . ; I; m ¼ 1; . . . ;M ð6:1Þ

And, using the lowered inputs for all originally-inefficient subunits, and the original inputs for all other subunits, computethe revised aggregated inputs for all agencies.

~Xi ¼X4

m¼1

~xim i ¼ 1; . . . ; I ð6:2Þ

Then, for the target agency, compute its revised efficiency score

mink

w ð6:3Þ

subject toXI

i¼1

~Xiki 6 w~Xk ð6:4Þ

XI

i¼1

Yiki P Yk ð6:5Þ


i¼1

ki ¼ 1 ð6:8Þ

If target k’s score is one or greater (w P 1), it is allocatively efficient when compared to its peer agencies.


5.6. Estimate resource reallocations that minimize target agency costs while maintaining output

The remaining task is to estimate improvements that would be possible if the target agency’s resources were optimallyreallocated among its subunits. In the previous tasks, the goal was either to maximize efficiency by choosing optimal weightsfor the various subunits or the DMU as a whole, or to estimate the effects of those efficiencies on overall agency performance.

Although the objective of maximizing DEA efficiency is appropriate when choosing virtual weights by which to multiplythe fixed amounts of inputs and outputs, efficiency-maximization should not be the objective of resource reallocation inmost cases (Gleason and Barnum, 1982). That is, the typical objective for reallocating resources would be either to maximizeoutput while holding inputs constant, or minimize inputs while holding output constant. Simply maximizing efficiency withno restrictions on input and output levels will not necessarily result in the same solution, as demonstrated by Beasley (2003)in which efficiency-maximization decreased both inputs and outputs.

The difference between the outcomes of efficiency-maximization and input-minimization-while-holding-output-con-stant is critical for urban transit. For example, transit agencies could generally increase their riders-per-bus-hour efficiencyratio by decreasing bus service hours, because, for every 1% decrease in hours of service, ridership decreases by only about0.5% on the average (Litman, 2004, 2007). Thus, for most transit systems, offering less service would increase the riders/bus-hour efficiency ratio but also would decrease ridership. So adjusting input to maximize efficiency would likely result in veryfew riders and very little service, which is not a desirable outcome. A much more likely objective would be to minimize costswhile holding service constant, which, as a collateral benefit, will also increase efficiency.

There is a large body of literature developing models that address the reallocation issue under a wide variety of circum-stances, with good reviews found in (Golany and Tamir, 1995; Lozano et al., 2004; Lozano and Villa, 2005; Nesterenko andZelenyuk, 2007; Cook and Seiford, 2009; Kao, 2009a), but none of them fit the very simple but very exact requirementsneeded here. In particular, we require a model that reallocates resources to minimize agency input while maintaining con-stant output, accepts the current efficiency levels of the subunits as a given (rather than assuming fully efficient subunits),considers the efficiency of each subunit relative only to other subunits of the same agency, reflects the unique functionalrelationships between each subunit’s input and output, and contains constraints representing political and technical reali-ties. A model that sets efficient targets for the individual subunits, and then attempts to increase the agency’s allocation effi-ciency assuming all subunits are efficient, would not likely provide a usable answer. It is unlikely that all subunits wouldactually become efficient and therefore the allocation decision would be incorrect. Unlike an earlier task in which we wishedto know how efficient each type of subunit was compared to its peers in other agencies, in this task we simply wish to real-locate resources based on how each subunit compares to other subunits in its agency. And, since there always will be polit-ical and technological constraints, it is necessary to make provision for these in the model if we wish our prescribedreallocation to be realistic.

Because of Task Six’s objectives and constraints, DEA cannot be applied. In this final task, therefore, our earlier DEA mod-els are replaced with a general mathematical programming model that does incorporate the required objectives and con-straints. This mathematical program may be either a linear or a nonlinear program, depending on the nature of therelationships among the variables.

We present our model in Mathematical Program 7, which reallocates resources among an agency’s subunits so that thetotal agency expenses will be lowered without decreasing total output. Changes in input values cause changes in the outputvalues, so it is not possible to reallocate the inputs without taking into account the effects on outputs. Unless we know thetrue functional relationships between inputs and outputs for each of the production processes, it is necessary to estimatethem using suitable methods.

minym xm

X4

m¼1

xm ð7:0Þ

subject toXM

m¼1

xm 6 i ð7:1Þ

X4

m¼1

ym PX4

m¼1

xm ð7:2Þ

ym P axm m ¼ 1;2;3;4 ð7:3ÞX2

m¼1

ym P bX2

m¼1

xm ð7:4Þ

xm P 0 m ¼ 1;2;3;4 ð7:5Þym ¼ f ðx1mÞ m ¼ 1;2;3;4 ð7:6Þ

In this illustration, there is only one type of input, which is to be allocated among the M = 4 production processes. Specifi-cally, m = 1 is self-operated paratransit service, m = 2 is outsourced paratransit service, m = 3 is self-operated motorbus ser-vice, and m = 4 is outsourced motorbus service. It is assumed that the target transit agency has two, three or four subunits, soreallocation is possible. Constraint 7.1 requires that the total of the reallocated funds not exceed their original value (i). Con-


straint set 7.2 requires that the total seat-hours (P4

m¼1xm) not decline. Constraint set 7.3 requires that each subunit retain atleast a percent of its original seat-hours xm, and constraint 7.4 requires that the combined paratransit operations retain atleast b percent of their total service b

P2m¼1xm. Constraint set 7.5 requires all subunit input amounts to be non-negative. Con-

straint set 7.6 estimates the new outputs ym produced by the new levels of input xm, with one estimation equation for each ofthe m outputs; linear programming can be used to solve the model if all relationships in constraint set 7.6 are linear, butnonlinear programming must be used otherwise.

The preceding constraints are illustrative of how the model can be adapted to the needs of the particular agency involved.The constraints could have different cut-off values, and there could be more, fewer or different constraints.

Further, an agency may wish to replace statistical estimates of changes in outputs resulting from changes in inputs, suchas those that we use in constraint set 7.6, with estimates of marginal effects based on their own internal analysis and data. Ifrecent contracts either increase or decrease the cost per seat mile of a particular type of service, then this must be reflected inthe forecasts. That is, while we use the relationships identified based on our empirical estimates of the parameters of Eq. (8)for the four service types, we would expect agencies applying the model to make use of the best data and judgments theyhave available.

6. Material

6.1. Data collection

This study uses one input and one output from each of the four subunits available to an urban transit agency, for a po-tential of four inputs and four outputs per agency. As already noted, the output variable is seat-hours, and the input variableis operating expenses adjusted for price differences over time and among urban areas. The sample consists of 2002–2006data from the National Transit Database for 52 transit agencies chosen from the approximately 62 agencies with 150 or morevehicles in maximum service (United States Federal Transit Administration, 1997–2008). We limited our sample to the larg-est agencies because only they are required to report all of the variables needed. Equally important, transit systems of rad-ically different sizes cannot be validly compared, as Karlaftis, McCarthy and Sinha have demonstrated with a variety ofdifferent samples and diverse statistical methodologies (Karlaftis and McCarthy, 1997; Karlaftis et al., 1999; Karlaftis andMcCarthy, 2002; Karlaftis, 2003, 2004, 2010). We include all of agencies that accurately reported all of the data neededfor all five years. Subunits include:

51 self-operated motorbus subunits,15 outsourced motorbus subunits,22 self-operated paratransit subunits,36 outsourced paratransit subunits.

6.2. Input measure

We use total operating expenses as our measure of input, as a proxy for physical inputs. Because we are using the US Na-tional Transit Database, the operating expenses of each subunit include vehicle operation and maintenance, non-vehiclemaintenance, and administration. Not included are any overall agency expenses incurred in support of subunit services,or depreciation. More information can be gained by consulting the National Transit Database accounting requirements (Uni-ted States Federal Transit Administration, 1997–2008).

It is more typical in transit DEA studies to use physical inputs, usually labor, fuel, and vehicles (De Borger et al., 2002).However, DEA assumes that there is substitutability among inputs, with diminishing marginal rates of substitution (Peter-sen, 1990). For transit, this would mean that, to produce a fixed level of output, a bus or paratransit operation could substi-tute labor for vehicles, or substitute vehicles for fuel. In truth, there is very little substitutability in this industry; inputs haveto be used in a virtually fixed ratio, with any excess being wasted. (This reasoning has been confirmed by Joseph DiJohn, Re-search Professor, Urban Transportation Center, University of Illinois at Chicago. From 1983 to 1998, Professor DiJohn wasChief Executive Officer of Pace, the Suburban Bus Division of the Illinois Regional Transportation Authority. Pace is respon-sible for all bus service in the six county region of Northeastern Illinois outside of Chicago.) Disaggregating nonsubstitutableinputs would usually result in some truly inefficient units being reported as efficient. As is well known, increasing the num-ber of inputs and outputs results in higher efficiency scores, whether or not they are justified. Also, the number of DMUsreporting slacks normally will increase, making the radial technical efficiency scores less meaningful. Further, because ofthe random error component in all variables, the larger the number of inputs and outputs, the more likely that a DMU’s effi-ciency score will be increased by random chance.

Further, it has recently been shown that conventional DEA models report invalid efficiency scores when outputs and/orinputs are nonsubstitutable, using real-world data from hospitals and bus transit (Barnum et al., 2010a; Barnum and Glea-son, 2010b). These studies demonstrate that DEA efficiency estimates are biased when inputs and outputs are nonsubstitut-able. The degree of bias varies considerably among decision making units, resulting in substantial differences in efficiencyrankings between DEA and the new measures. And, most of the units that DEA identifies as efficient are, in truth, not effi-


cient. In short, when inputs and outputs are not substituted for either technological or socio-economic/legal reasons, it isinvalid to simultaneously include such variables in conventional DEA models.

Of course, using total resource expense as a proxy for physical resource inputs requires that the expenses be adjusted forchanges in prices over time and differences in prices across cities. The mean operator platform-hour wage rate for self-oper-ated bus subunits is used as a proxy for differences in the price of input resources caused by inflation over time, and cost ofliving differences among cities.

6.3. Output measure

We use estimated seat-hours as our output variable, because vehicle hours would be an inappropriate comparative mea-sure. The typical motorbus not only is much more expensive, but also has many more seats than paratransit vehicles, andmotorbuses typically carry many more passengers per vehicle, with paratransit vehicles seldom carrying more than oneor two passengers at a time. There are however a great variety of sizes for paratransit vehicles ranging from automobilesto mini-buses. Motorbuses themselves range in size somewhat, with about half of the buses delivered over the past decadehaving 40 or more seats and most of the rest having 30–39 seats (American Public Transit Association, 2010a,b). Putting all ofthe information together, we assume that the ratio in seating capacity between the typical motorbus and the typical para-transit vehicle is about 10 to 1. We therefore estimate seat-hour data by multiplying motorbus vehicle hours by 40 and para-transit vehicle hours by 4.

Others using the procedures we suggest in this article could substitute their own preferences for the methodology formeasuring seat-hours, or, indeed, for the type output or outputs desired. Neither would affect the underlying procedurein any way. For example, a more precise measure of seat-hours would be to multiply the number of actual hours that eachindividual vehicle was on the street by the actual seats available in that vehicle, and aggregate the products over all vehiclesin the subunit involved. The aggregated totals for each subunit would be used in place of the averages we employ herein.

Or, one might prefer to use seat-miles, passenger-trips, passenger-miles or passenger-hours (among many other possibil-ities) as the output variables. If multiple outputs are utilized, they must be substitutes for each other in order to obtain validefficiency scores from conventional DEA models (Barnum et al., 2010a; Barnum and Gleason, 2010b). So, it generally wouldbe invalid to simultaneously employ proxies for service supplied and service consumed because they are complements, notsubstitutes – increases in service supplied results in increases in service consumed (Litman, 2004, 2007).

We chose to use a proxy for service supplied (seat-hours) for several reasons. First, production is much more under thecontrol of management than consumption, so it is much more likely that management can act to institute improvements.Second, service supplied is a pure efficiency indicator, while service consumed is often considered more of an effectivenessmeasure. Third, it has been very common in transit efficiency studies to use proxies for service supplied as the output. Fi-nally, Litman has demonstrated that there is a relatively close relationship between production and consumption, so, tosome degree service supplied is a proxy for service consumed if everything else is held equal. There has been a great dealof controversy about the value of utilizing production vs consumption over the years (Karlaftis, 2004), but we do not takea stand on the issue herein. We have chosen to use a proxy for supply to illustrate the procedure, but exactly the same pro-cedure would be used if the outputs were proxies for consumption.

6.4. Estimation of functional relationships between inputs and outputs

Because estimation of functional relationships between inputs and outputs is the same for all agencies, we illustrate itonce at this point. We must determine whether the appropriate DEA model involves returns to scale that are constant, var-iable with output increasing at a decreasing rate, or variable with output increasing at an increasing rate. The standard meth-od of making the decision is to regress each output on its own inputs and statistically test the regression parameters forlinearity (Coelli et al., 2007). We employ a fixed-effects, robust regression model of seat-hours on adjusted operating ex-penses and adjusted operating expenses squared, with 5 years of data (Eq. (8)).

sjt ¼ aj þ b1ej þ b2e2j þ ujt ð8Þ

A separate regression is conducted for each of the four modes and for the aggregated values of all four modes, so there are atotal of five regressions, with the variable values being those for one of the four modes or the aggregated values for the sys-tems as a whole. So, for each one of the four modes or for their aggregated value, sjt is the seat-hours from system j and timeperiod t, aj is the individual effect of system j, ej is the operating expenses from system j, and ujt is the random error in theresponse variable sjt from system j in period t. If the regression coefficient b2 does not differ from zero to a statistically sig-nificant degree, than the relationship is linear. If b2 is negative with statistical significance, then seat-miles are increasing at adecreasing rate. Table 1 shows results for the aggregated agency and each of the four subunits.

All estimations reported quadratic relationships with the linear term positive and the quadratic term negative to statis-tically significant degrees. These findings dictate use of conventional variable-returns-to-scale DEA models (Linear Programs1 and 2). Please note that they also require the use of nonlinear programming for Mathematical Program 7 because equationset 7.6 contains nonlinear constraints; we employed a standard GRG nonlinear program, using Premium Solver Platform forExcel, Version 8.

Table 1Fixed-effects, robust regression of seat-hours on adjusted expenses, 2002–2006 data.

Seat-hours (Expenses) (Expenses)2 � 10�8 Constant

Agency 0.0580451 �0.0124642 1.06e+07Self-operated paratransit 0.1007782 �0.2020366 �8891.144Outsourced paratransit 0.1043372 �0.0124718 �217763.3Self-operated motorbus 0.0663095 �0.0021405 8864466Outsourced motorbus 0.153783 �0.0281704 374951.7

Note: All regression coefficients statistically significant at the 0.01 level except for that of self-operated paratransitexpenses squared, which was statistically significant at the 0.027 level.


7. Results

The procedure is designed for a comprehensive audit of transit efficiency in a single metropolitan area, not for analysis ofall systems in the sample. However, to make the process more clear, we illustrate the procedure with three different metro-politan transit agencies: Maryland Transit Administration (MTA), Transit Authority of River City Kentucky (TARC), and theRhode Island Public Transit Authority (RIPTA). MTA and TARC have four subunits each, and RIPTA has three (Table 2). Weselected these three agencies because they serve relatively typical, mid-sized metropolitan areas; the very biggest cities havecomplexities that would make the application less transparent and the very smallest cities aren’t complex enough to dem-onstrate the full range of the procedure.

We use 2002–2006 panel data to estimate functional relationships among the variables (Table 1), but only 2006 data tocompute DEA scores. Although it would have been possible to use all five years of data to estimate efficiency trends or 5-yearmean efficiencies, our focus in this paper is on the new methodology, not on intertemporal analysis, so we chose to use onlythe most recent year for the DEA computations.

We assume, for all input resource reallocations, that each subunit must maintain at least 90% of its seat-hour output andparatransit as a whole must maintain at least 95% of its seat-hour output; it is not reasonable to allow large changes becausethey might be beyond valid range of our forecasting equations (Table 1).

Please carefully note that we use the forecasting equations in Table 1 for Task 6 because they represent the best informa-tion that we have about the effects of changes in resource allocations for a particular system. The team of transit profession-als and analysts for a particular system would have access to all of that system’s internal data, so they might well be able toaccurately predict the effects of larger changes than the 5–10% that we have restricted our model to. Indeed, the main valueof allowing small resource allocations is that the results point to the directions of reallocations that would increase efficiencywhile remaining within a valid prediction range, and identifying the optimal directions for reallocations may be all that isnecessary.

Because we are allowing only small proportional changes in resource allocation, and these are even more restricted be-cause the initial amounts of resources going to those subunits that lose resources are small, the effect on expenses will alsobe relatively minor. However, as demonstrated in the case studies, the ratio of the percentage decrease in expenses to per-centage reallocation of seat-hours is large (reallocation elasticity), which is the really relevant measure.

The results of our analyses are presented in Tables 3 and 4, and discussed below.

7.1. The Maryland Transit Authority (MTA)

The overall (technical and allocation) efficiency of the MTA was 0.66, with subunit technical efficiencies of 0.29 for self-operated paratransit (lowering overall agency efficiency by 0.02), 0.63 for outsourced paratransit (lowering overall agencyefficiency by 0.03), 0.70 for self-operated motorbuses (lowering overall agency efficiency by 0.18), and 0.23 for outsourcedmotorbuses (lowering overall agency efficiency by 0.06). Thus, its overall efficiency would be helped most by improving theself-operated motorbus subunit. Its secondary targets for improvement would be the outsourced bus service and the self-operated demand responsive service, which have extremely low efficiency ratings compared to their peers at other agencies.

Table 2Vehicles in maximum service, illustrative agencies, 2006 data.

Agency Paratransit Motorbus

Self-operated Outsourced Self-operated Outsourced

Maryland Transit Administration 41 206 508 168Rhode Island Public Transit Authority 86 33 203 0Transit Authority of River City 4 78 182 7

Source: United States Federal Transit Administration, 1997.

Table 3Illustrative agencies, Tasks 2–5.

Agency Task 3: Agency efficiency Task 2: Subunit efficiency

Self-operated paratransit Outsourced paratransit Self-operated motorbus Outsourced motorbus

MTA 0.66 0.29 0.63 0.70 0.23TARC 0.84 2.98 0.77 0.91 0.91RIPTA 0.75 1.11 0.70 0.81 na

Task 4: Change in aggregated efficiency when target subunit efficiency = 1MTA 0.02 0.03 0.18 0.06TARC �0.02 0.03 0.07 0.00RIPTA �0.01 0.01 0.14 na

Task 5: Allocation efficiencyMTA 0.98TARC 0.81RIPTA 0.77

Table 4Recommended changes in adjusted operating expense allocations (Task 6).

Agency Self-operated Outsourced Self-operated Outsourced TotalParatransit Paratransit Motorbus Motorbus

MTAa

Original expenses 12,163,624 27,553,779 188,828,975 30,110,289 258,656,666Original seat-hours 419,476 2636,556 73,361,240 10,575,520 86,992,792Recommend expenses 9751,748 26,268,921 192,021,406 25,178,632 253,220,707Estimated seat-hours 377,528 2525,702 74,571,594 9517,968 86,992,792Change in expenses ($) �2411,876 �1284,858 3192,431 �4931,657 �5435,959Change in expenses (%) �19.83 �4.66 1.69 �16.38 �2.10Change in seat-hours (h) �41,948 �110,854 1210,354 �1,057,552 0Change in seat-hours (%) �10.00 �4.20 1.65 �10.00 0Original efficiency 65.51%New efficiency 66.92%

TARCa

Original expenses 625,231 9056,075 49,521,677 660,487 59,863,470Original seat-hours 31,172 1048,116 24,971,280 623,000 26,673,568Recommend expenses 538,192 7832,671 49,698,278 558,982 58,628,124Estimated seat-hours 28,055 997,269 25,087,544 560,700 26,673,568Change in expenses ($) �87,039 �1223,404 176,601 �101,505 �1,235,346Change in expenses (%) �13.92 �13.51 0.36 �15.37 �2.06Change in seat-hours (h) �3117 �50,847 116,264 �62,300 0Change in seat-hours (%) �10.00 �4.85 0.47 �10.00 0Original efficiency 84.46%New efficiency 86.14%

R1PTAa

Original expenses 7767,869 3,948,974 60,177,807 0 71,894,650Original seat-hours 993,684 390,268 26,940,280 0 28,324,232Recommend expenses 6391,042 4241,000 60,449,265 0 71,081,307Estimated seat-hours 894,316 420,439 27,009,478 0 28,324,232Change in expenses ($) �1376,827 292,026 271,458 0 �813,343Change in expenses (%) �17.72 7.39 0.45 �1.13Change in seat-hours (h) �99,368 30,171 69,198 0 0Change in seat-hours (%) �10.00 7.73 0.26 0Original efficiency 74.66%New efficiency 75.51%

a Each subunit must maintain at least 90% of its original seat-hours, and paratransit as a whole must maintain at least 95% of its scat-hours.


The MTA was the most efficient of the three agencies in allocating its resources to its four subunits, as evidenced by itsallocation efficiency of 0.98. Nevertheless, we apply Mathematical Program 7 to subunit inputs and outputs. As can be seenin Table 4, the results suggest that self-operated paratransit seat-hours be decreased by 10% and outsourced paratransit seat-hours be decreased by 4%, for an overall decrease in paratransit seat-miles of 5%. This indicates, not surprisingly, that it ismore efficient to carry passengers by motorbus, and that of the two paratransit operations, outsourcing is more efficient.Further, the agency will become more efficient if outsourced motorbus operations are decreased by 10% and self-operatedmotorbus operations are increased by 2%, with the increase coming partly from paratransit and partly from outsourcedmotorbuses. The reason for the switch in service from outsourced to self-operated motorbuses is likely the result of the largenumber of seat-hours produced, which allows the in-house unit to have relatively low marginal costs compared to the sub-


contractor. Also, the extremely low efficiency of the subcontracted bus service likely has affected the two bus subunits’ com-parative performance.

The reallocation would save $5.4 million or about 2.1% of the MTA’s budget. Because 2.9% of the total seat-hours werereallocated, for every 1% of seat-hours reallocated, operating expenses decline by about 0.8%.

7.2. The Transit Authority of River City (TARC)

TARC has the highest overall technical and allocation efficiency of the three agencies at 84%, partly because its superef-ficient self-operated paratransit subunit compensates for its relatively inefficient outsourced paratransit unit. Its two motor-bus subunits are equally efficient compared to their peers at 91%. To improve overall efficiency by improving subunitefficiency, TARC should focus on its self-operated motorbus subunit, which, if it becomes efficient, will increase TARC’s effi-ciency by 0.07.

Although TARC had the highest aggregated efficiency of the three systems considered, it has a low allocation efficiency of0.81, pointing to possible resource misallocations. Applying the reallocation program yields the following recommendationsfor allocation efficiency improvement. Therefore, the maximum allowable seat-hours should be shifted from paratransit tomotorbus, with as many of the seat-hours as possible coming from the in-house paratransit subunit. And, it is recommendedthat the paratransit seat-hours as well as 3% of the outsourced motorbus seat-hours be provided by the self-operated motor-bus subunit.

Recommended reallocations would save 2.1% of the agency’s 2006 budget, or about $1.2 million. Because only 0.9% of thetotal seat-hours were reallocated, for every 1% of seat-hours reallocated, operating costs decline by about 2.4%.

7.3. The Rhode Island Public Transit Authority (RIPTA)

RIPTA’s overall technical and allocation efficiency was 0.75. Self-operated paratransit had a technical efficiency of 1.11(raising agency efficiency by 0.01), outsourced paratransit technical efficiency was 0.70 (lowering overall agency efficiencyby 0.01), and self-operated motorbus efficiency was 0.81 (lowering overall efficiency by 0.14). As usual, the biggest increasein RIPTA efficiency could be made by improving the self-operated motorbus subunit. RIPTA was the least efficient of thethree agencies in allocating its resources to its subunits, as evidenced by its allocation efficiency of 0.77.

We apply the reallocation mathematical program to RIPTA’s three subunits’ inputs and outputs, making the commonassumptions that each of the three must maintain 90% of their original output and paratransit must maintain 95% of its ori-ginal output. As before, the 5% of paratransit seats available for switching are switched to the only motorbus subunit. In RIP-TA’s case, unlike the preceding two, most paratransit service is provided by its in-house subunit. Because the outhousesubunit is more efficient, the in-house unit loses the maximum 10% of its seat-hours, but the outhouse unit gains 8% moreseat-hours, a net loss of 5% of the total seat-hours which are switched to buses.

This would result in a savings of an estimated $813,000 or 1.1% of RIPTA’s 2006 budget. Because 0.9% of the total seat-hours were reallocated, for every 1% of seat-hours reallocated, operating expenses declined by about 1.6%.

8. Discussion

In order to improve a multi-subunit transit agency’s efficiency, it first is necessary to validly measure all key aspects ofthat efficiency. Developing a methodology to do so is the sole purpose of this paper.

After these performance indicators are collected, it is necessary to identify the causes behind their values and determinewhat actions, if any, can be taken to improve them. Illustrating this process is beyond the scope of this already-long paper.However, to provide usable, valid information about the efficiency of a given metropolitan area’s transit system, the raw effi-ciency scores must be subjected to detailed, rigorous analysis, and adjusted as necessary. It is necessary to sort out whichscores can be improved by managerial actions and reallocations, and which elements of the scores are caused by environ-mental factors such as political-social-legal-economic-financial constraints, mainline vs. feeder service, interactions amongsubunits, age and technology of the system, population size, density, and geographic and income distributions, and manymore. Estimations from the sample data are all we employ in this demonstration. Such generalizations are useful input,but they certainly are not sufficient.

In short, before the efficiency scores can be validly used to either evaluate a certain system’s or subunit’s performance, makechanges in attempts to improve it, or reallocate resources, they must be subjected to a rigorous joint examination by quanti-tative data analysts and transit professionals intimately familiar with the particular agency and its subunits. Indeed, even inour development in this paper of the general methodology and performance indicators, it was not by chance that our teamincludes coauthors with substantial transit industry professional experience and coauthors with quantitative acumen.

Our algorithm (Fig. 2) can easily be expanded to include multiple inputs and multiple outputs. We don’t do so here be-cause they are not valid for transit analysis. Further, the extra subscripts and summations required would make our math-ematical models more opaque and therefore more difficult to intuitively understand.

In order to analyze the efficiency with which public transportation is provided in an urban area, it is insufficient to eval-uate only aggregated performance, because the reasons for that performance are found in an agency’s subunits. It also is


insufficient to evaluate only some of the subunits, because this results in an incomplete understanding of the reasons for anagency’s overall efficiency. Further, individual subunit efficiency is mainly important because of its effect on the entire sys-tem. So, it is necessary to identify the impact on agency efficiency of changes in each subunit’s efficiency.

Finally, because policy decisions concerning the allocation of resources among the subunits also affect aggregated agencyefficiency, the agency’s allocation efficiency must to be taken into account. In short, the overall efficiency of a metropolitanarea’s public transportation can be badly biased upward if allocation efficiency is ignored, and the only way to identify allo-cation efficiency is to simultaneously evaluate all subunits.

This last point, the necessity of identifying allocation efficiency, is graphically illustrated by the efficiency estimates forRIPTA (Table 3). RIPTA’s three subunits and their technical efficiencies are self-operated paratransit (1.11), self-operatedmotorbus (0.81) and subcontracted paratransit (0.70), and the mean of these three efficiencies is 0.87. Whether we weightby vehicles, seat-hours or operating expenses, the weighted mean technical efficiency of the three subunits is in the mid-.80s. Yet, the overall efficiency of RIPTA is only 0.75. The 10–12% decline in total efficiency when compared the mean tech-nical efficiency is caused by RIPTA’s allocation inefficiency. RIPTA’s allocation efficiency is only 0.77. Since total efficiencyencompasses the effects of the technical efficiencies of the individual subunits and the efficiency with which the agency allo-cates resources to the subunits, it is not surprising to total efficiency is so much lower than mean technical efficiency.

On the bright side, the lower an agency’s allocation efficiency, the higher the savings for every 1% change in the reallo-cation of seat-hours (reallocation elasticity). Both RIPTA and the MTA had four subunits, so these two are directly compara-ble. While RIPTA’s allocation efficiency was only 0.77, MTA’s was 0.98. For every one percent change in the allocation of seat-hours, RIPTA would lower its operating expenses by 2.5% while MTA would lower its operating expenses by only 0.76%, onlyone-third as much.

One may wish to note that the reallocations all increased the overall efficiency scores, but not by much. This was mainlydue to the constraints on the amount of reallocation that could occur, and partly due to the fact the the reallocation math-ematical program’s objective was to minimize expenses while holding output constant, rather than the DEA objective ofmaximizing efficiency (Beasley, 2003).

This new transit efficiency evaluation procedure provides, for the first time, comprehensive efficiency information forboth urban transit policy and for transit management practice. It informs policy by integrating the effects of the varietyof subunits making up a city’s public transportation system, and identifying the effects of each subunit on the overall results.It further informs policy by making it possible to identify resource reallocations among subunits that will improve overallperformance. It informs managerial practice by identifying the efficiency with which each major subunit type provides tran-sit service, so management of each subunit will know how their efficiency compares to the same type of subunit in othercities. Further, because many aspects of the procedure are new to the DEA literature, it contributes to the body of knowledgeof DEA analysis of organizations with multiple subunits.

9. Conclusions

This paper presents a Data Envelopment Analysis (DEA) protocol for analyzing the efficiency of metropolitan transitagencies that oversee multiple types of transportation service. The protocol is illustrated by applying it to US transit agen-cies that can serve their cities with four types of subunits: self-operated motorbus, outsourced motorbus, self-operatedparatransit, and outsourced paratransit. DEA scores that are estimated for a target agency include: (1) overall efficiencyof the target agency as a whole, (2) technical efficiency of each of the target agency’s subunits when each subunit is com-pared only to other subunits of the same type, (3) effect of changes in each of service type’s efficiency on its parent agency’sefficiency and (4) the agency’s allocation efficiency in apportioning resources among its subunits. Finally, a mathematicalprogramming algorithm is illustrated that allocates target agency resources to its subunits with the objective of decreasingthe cost of transit in the urban area while holding total service constant. The protocol thereby provides a comprehensiveaudit of a target transit agency’s efficiency in supplying service to its metropolitan area, including recommendations forimprovement.

This paper contributes to the literature by comprehensively addressing metropolitan area transit efficiency issues thathave heretofore been ignored or done incorrectly. It is the first to validly identify the individual technical efficiencies of eachtype of transit employed in an urban area, to identify allocation and technical efficiency in the DEA aggregated score for alltransit types, to estimate the effect of changes in each type of transit’s efficiency on the overall public transportation effi-ciency, to estimate the efficiency of the metropolitan area’s allocation of funds among its transit modes, and to offer a meth-od for reallocating resources in order to improve efficiency.

Acknowledgements

Many significant changes were made at the suggestion of two anonymous referees. These changes have greatly improvedthe quality and credibility of this paper.

Table A1Efficiency and seat-hours by agency and subunit, 2006 data.

ID Total Technical efficiency Total seat-hours

Efficiency Paratransit Paratransit Motorbus Motorbus Seat-hours Paratransit Paratransit Motorbus MotorbusAgency Self-operated Outsourced Self-operated Outsourced Agency Self-operated Outsourced Self-operated Outsourced

1 0.84 0.52 0.89 120,717,992 3,209,272 117,508,7203 0.77 0.49 0.53 0.89 24,931,644 239,148 626,296 24,066,2008 0.76 0.58 0.79 80,881,456 2,111,016 78,770,440

1001 0.75 1.11 0.70 0.81 28,324,232 993,684 390,268 26,940,2801003 1.01 1.33 1.06 0.65 117,142,212 4,907,732 107,085,200 5,149,2801048 1.02 0.95 21,226,840 21,226,8402004 0.77 0.53 0.77 33,925,432 262,992 33,662,4402007 0.73 0.78 0.75 37,936,780 957,420 36,979,3602008 1.00 1.00 1.00 620,187,932 8,336,172 611,831,7602113 0.82 0.63 0.86 173,982,348 402,588 4,255,348 20,124,3603019 0.86 0.68 0.90 168,991,556 4,768,876 169,727,0003030 0.92 0.80 0.95 0.50 86,992,792 419,476 2,636,556 159,832,080 4,390,6003034 0.66 0.29 0.63 0.70 0.23 19,362,912 523,792 73,361,240 10,575,5204003 0.86 0.90.591 0.88 35,028,800 635,400 18,839,1204008 0.84 2.98 0.87 0.54 26,673,568 31,172 1,048,116 33,004,880 1,388,5204018 0.84 1.16 0.77 0.91 0.91 79,712,848 1,164,408 24,971,280 623,0004022 1.04 1.04 52,633,752 2,538,552 78,548,4404029 1.01 1.05 0.95 1.98 43,147,216 1,795,376 41,413,560 8,681,6404035 0.84 0.71 0.69 0.94 24,549,440 212,240 41,351,8404041 0.72 0.69 0.71 31,034,016 408,096 24,337,2004086 0.76 0.72 0.77 16,697,984 128,592 373,512 30,625,9205005 0.93 0.66 1.00 58,396,416 1,623,496 16,195,8805008 0.87 0.61 0.93 37,092,028 630,588 56,772,9205012 0.82 0.36 0.53 0.86 73,589,604 677,756 258,208 36,461,4405015 0.83 0.80 0.86 25,458,712 499,232 16,195,8805016 0.80 0.66 0.81 14,260,932 345,852 24,337,2005022 1.18 0.74 1.03 85,405,120 30,625,9205027 0.91 0.37 0.86 33,045,120 717,684 311,152 36,461,4405031 0.62 0.72 0.53 0.70 0.61 7737,948 1,148,028 72,653,640 1,144,7605032 1.00 1.70 276,765,580 2,401,340 24,956,4805066 0.97 0.68 1.02 70,805,776 5,431,496 85,405,1205113 0.74 0.92 0.95 0.29 51,228,752 617,352 30,871,560 8,325,8405119 0.45 0.47 0.44 125,587,092 3,299,172 6589,9206008 0.75 0.58 0.59 0.70 0.52 57,600,788 1,104,816 984,132 274,364,240 25,007,7206011 0.84 0.65 0.96 87,091,000 57,048,4406056 0.78 0.85 0.73 29,059,104 205,620 806,684 520,671,4007005 0.85 1.00 0.97 0.88 56,089,460 1,512,580 97,280,2007006 0.80 0.53 0.83 48,356,644 714,476 467,128 55,511,8408001 0.76 0.46 0.73 1.92 127,660,960 2,573,400 47,184,0408006 1.00 1.01 0.56 0.88 60,368,235 1,493,264 27,891 65,903,560 59,184,0009002 0.82 2.58 0.74 0.36 32,609,640 880,600 58,847,0809009 0.80 0.65 0.70 35,504,476 1,991,756 23,181,040 8,548,0009013 0.65 0.57 0.54 1.49 15,236,000 245,440 53,512,7209016 0.57 0.60 0.74 31,198,972 754,612 14,723,880 263,6809019 0.70 0.51 0.94 28,420,812 49,292 30,444,3609023 0.97 0.51 0.79 36,095,240 28,371,5209026 0.83 0.78 21,221,176 284,736 36,095,2409030 0.77 0.51 61,380,428 1,602,948 20,936,4409032 0.71 0.48 1.00 24,453,336 1,137,936 59,777,4809033 0.78 0.75 0.91 85,027,408 2,720,728 23,315,4009036 1.75 0.57 0.75 0.62 333,007,760 76,715,520 5,591,1609154 1.07 0.99 0.68 311,651,720 21,356,040

Note: ID is Federal Transit Administration identification number.


Appendix A

See Table A1.

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improving the efficiency of metropolitan area transit by joint analysis of its multiple providers

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