Measuring and Managing the Productivity of U.S. Public Transit Systems: An Unoriented Network DEA

Abstract

Governments at all levels face budget shortfalls, and consequently public transit systems in the U.S. must compete with other public services for financial support. In order to depend less on public funding, it is critical that public transit systems focus on their operational performance and identify any sources of inefficiency. In this chapter, we present an unoriented network DEA methodology that measures a public transit system’s performance relative to its peer systems, compares its performance to an appropriate efficient benchmark system, and identifies the sources of its inefficiency.

Appendix

1.1 Iterative Algorithm for the Commuter Rail Unoriented Network DEA Model

To measure the overall commuter rail system efficiency, we apply the innovative iterative process described in Lewis et al. (2013) and Holod and Lewis (2011) to commuter rail systems in each revenue year separately.

Define,

• ε _1kt and ε _4kt to be the efficiencies of the first and fourth stage models for commuter rail system k during iteration t, respectively

• ε _2ktf and ε _3ktf to be the efficiencies of the second and third stage models for commuter rail system k during the forward pass of iteration t, respectively

• ε _2ktb and ε _3ktb to be the efficiencies of the second and third stage models for commuter rail system k during the backward pass of iteration t, respectively

• θ _1kt and θ _4kt to be the approximate inverse efficiencies of the first and fourth stage models for commuter rail system k during iteration t, respectively

• θ _2ktf and θ _3ktf to be the approximate inverse efficiencies of the second and third stage models for commuter rail system k during the forward pass of iteration t, respectively

• θ _2ktb and θ _3ktb to be the approximate inverse efficiencies of the second and third stage models for commuter rail system k during the backward pass of iteration t, respectively

• λ _jt to be the weight placed on commuter rail system j by commuter rail system k when solving the first stage model during iteration t

• ω _jtf and ω _jtb to be the weights placed on commuter rail system j by commuter rail system k when solving the second stage model during the forward pass and the backward pass of iteration t, respectively

• δ _jtf and δ _jtb to be the weights placed on commuter rail system j by commuter rail system k when solving the third stage model during the forward pass and the backward pass of iteration t, respectively

• μ _jt to be the weight placed on commuter rail system j by commuter rail system k when solving the fourth stage model during iteration t

• λ ^* _jt , ω ^* _jtf , ω ^* _jtb , δ ^* _jtf , δ ^* _jtb , and μ ^* _jt denote the optimal weights

We formulate the first stage VRS model for commuter rail system k during iteration t as follows:

$$ \begin{array}{cc}\hfill \mathrm{Min}\ {\varepsilon}_{1 kt}\ \mathrm{or}\ \mathrm{Max}\ {\theta}_{1 kt}\hfill & \hfill (34)\hfill \\ {}\mathrm{subject}\ \mathrm{to}\hfill & \hfill \hfill \\ {}\hfill {\displaystyle \sum_{j=1}^n{\lambda}_{j t}}{(OP)}_{j0}\le \left\{\begin{array}{c}\hfill {\varepsilon}_{1 kt}{(OP)}_{k0}\kern1em t=1\hfill \\ {}\hfill {\varepsilon}_{1 kt}{\displaystyle \sum_{j=1}^n{\lambda}_{j t-1}^{*}{(OP)}_{j0}\kern1em t>1}\hfill \end{array}\right.\hfill & \hfill (35)\hfill \\ {}\hfill {\displaystyle \sum_{j=1}^n{\lambda}_{j t}}{(VM)}_{j0}\ge \left\{\begin{array}{c}\hfill {\theta}_{1 kt}{(VM)}_{k0}\kern1em t=1\hfill \\ {}\hfill {\theta}_{1 kt}{\displaystyle \sum_{j=1}^n{\omega}_{j\left( t-1\right) b}^{*}{(VM)}_{j0}\kern1em t>1}\hfill \end{array}\right.\hfill & \hfill (36)\hfill \\ {}\hfill {\displaystyle \sum_{j=1}^n{\lambda}_{j t}}=1\hfill & \hfill (37)\hfill \\ {}\hfill {\varepsilon}_{1 kt}+{\theta}_{1 kt}=2\hfill & \hfill (38)\hfill \\ {}\hfill {\lambda}_{j t}\ge 0\kern2.25em j=1,2,\dots, n\hfill & \hfill (39)\hfill \\ {}\hfill {\varepsilon}_{1 kt},{\theta}_{1 kt}\ge 0\hfill & \hfill (40)\hfill \end{array} $$

The objective function (Eq. 34) minimizes the relative efficiency of the first stage model of commuter rail system k during iteration t or equivalently maximizes its approximate inverse efficiency. The first constraint (Eq. 35) ensures that the hypothetical target commuter rail system for commuter rail system k during iteration t consumes no more of input (OP) than it does during iteration t − 1. The second constraint (Eq. 36) ensures that the hypothetical target commuter rail system for commuter rail system k during iteration t generates at least as much of the intermediate product (VM) as is consumed by the second stage during the backward pass of iteration t − 1.

We formulate the second stage VRS model for commuter rail system k during the forward pass of iteration t as follows:

$$ \begin{array}{ll}\mathrm{Min}\ {\varepsilon}_{2 ktf}\ \mathrm{or}\ \mathrm{Max}\ {\theta}_{2 ktf}\hfill & (41)\hfill \\ {}\mathrm{subject}\ \mathrm{to}\hfill & \hfill \\ {}{\displaystyle \sum_{j=1}^n{\omega}_{j t f}}{(VM)}_{j0}\le {\varepsilon}_{2 ktf}{\displaystyle \sum_{j=1}^n{\lambda}_{j t}^{*}}{(VM)}_{j0}\hfill & (42)\hfill \\ {}{\displaystyle \sum_{j=1}^n{\omega}_{j t f}}{(RM)}_{j0}\ge \left\{\begin{array}{l}{\theta}_{2 ktf}{(RM)}_{k0}\kern1em t=1\hfill \\ {}{\theta}_{2 ktf}{\displaystyle \sum_{j=1}^n{\omega}_{j\left( t-1\right) b}^{*}{(RM)}_{j0}\kern1em t>1}\hfill \end{array}\right.\hfill & (43)\hfill \\ {}{\displaystyle \sum_{j=1}^n{\omega}_{j t f}}{(VA)}_{j0}\le {(VA)}_{k0}\hfill & (44)\hfill \\ {}{\displaystyle \sum_{j=1}^n{\omega}_{j t f}}{(DRM)}_{j0}\le {(DRM)}_{k0}\hfill & (45)\hfill \\ {}{\displaystyle \sum_{j=1}^n{\omega}_{j t f}}=1\hfill & (46)\hfill \\ {}{\varepsilon}_{2 ktf}+{\theta}_{2 ktf}=2\hfill & (47)\hfill \\ {}{\omega}_{j t f}\ge 0\kern2.25em j=1,2,\dots, n\hfill & (48)\hfill \\ {}{\varepsilon}_{2 ktf},{\theta}_{2 ktf}\ge 0\hfill & (49)\hfill \end{array} $$

The objective function (Eq. 41) minimizes the relative efficiency of the second stage model of commuter rail system k during the forward pass of iteration t or equivalently maximizes its approximate inverse efficiency. The first constraint (Eq. 42) ensures that the hypothetical target commuter rail system for commuter rail system k during the forward pass of iteration t consumes no more of the intermediate product (VM) than is generated by the first stage during iteration t. The second constraint (Eq. 43) ensures that the hypothetical target commuter rail system for commuter rail system k during the forward pass of iteration t produces at least as much of the intermediate product (RM) as is produced by the second stage during the backward pass of iteration t − 1. The third and fourth constraints (Eqs. 44 and 45) ensure that the hypothetical target commuter rail system for commuter rail system k during the forward pass of iteration t operates under conditions with no more of the site characteristics (VA and DRM) than commuter rail system k.

We formulate the third stage VRS model for commuter rail system k during the forward pass of iteration t as follows:

$$ \begin{array}{ll}\mathrm{Min}\ {\varepsilon}_{3 ktf}\ \mathrm{or}\ \mathrm{Max}\ {\theta}_{3 ktf}\hfill & (50)\hfill \\ {}\mathrm{subject}\ \mathrm{to}\hfill & \hfill \\ {}{\displaystyle \sum_{j=1}^n{\delta}_{j t f}{(RM)}_{j0}}\le {\varepsilon}_{3 ktf}{\displaystyle \sum_{j=1}^n{\omega}_{j t f}^{*}}{(RM)}_{j0}\hfill & (51)\hfill \\ {}{\displaystyle \sum_{j=1}^n{\delta}_{j t f}}{(PM)}_{j0}\ge \left\{\begin{array}{l}{\theta}_{3 ktf}{(PM)}_{k0}\kern1em t=1\hfill \\ {}{\theta}_{3 ktf}{\displaystyle \sum_{j=1}^n{\delta}_{j\left( t-1\right) b}^{*}{(PM)}_{j0}\kern1em t>1}\hfill \end{array}\right.\hfill & (52)\hfill \\ {}{\displaystyle \sum_{j=1}^n{\delta}_{j t f}}{(PD)}_{j0}\le {(PD)}_{k0}\hfill & (53)\hfill \\ {}{\displaystyle \sum_{j=1}^n{\delta}_{j t f}}{(ST)}_{j0}\le {(ST)}_{k0}\hfill & (54)\hfill \\ {}{\displaystyle \sum_{j=1}^n{\delta}_{j t f}}=1\hfill & (55)\hfill \\ {}{\varepsilon}_{3 ktf}+{\theta}_{3 ktf}=2\hfill & (56)\hfill \\ {}{\delta}_{j t f}\ge 0\kern2.75em j=1,2,\dots, n\hfill & (57)\hfill \\ {}{\varepsilon}_{3 ktf},{\theta}_{3 ktf}\ge 0\hfill & (58)\hfill \end{array} $$

The objective function (Eq. 50) minimizes the relative efficiency of the third stage model of commuter rail system k during the forward pass of iteration t or equivalently maximizes its approximate inverse efficiency. The first constraint (Eq. 51) ensures that the hypothetical target commuter rail system for commuter rail system k during the forward pass of iteration t consumes no more of the intermediate product (RM) than is generated by the second stage during the forward pass of iteration t. The second constraint (Eq. 52) ensures that the hypothetical target commuter rail system for commuter rail system k during the forward pass of iteration k produces at least as much of the intermediate product (PM) as is generated by the third stage operation during the backward pass of iteration t − 1. The third and fourth constraints (Eqs. 53 and 54) ensure that the hypothetical target commuter rail system for commuter rail system k during the forward pass of iteration t operates under conditions with no more of the site characteristics (PD and ST) than commuter rail system k.

We formulate the fourth stage VRS model for commuter rail system k during iteration t as follows:

$$ \begin{array}{ll}\mathrm{Min}\ {\varepsilon}_{4 kt}\ \mathrm{or}\ \mathrm{Max}\ {\theta}_{4 kt}\hfill & (59)\hfill \\ {}\mathrm{subject}\ \mathrm{to}\hfill & \hfill \\ {}{\displaystyle \sum_{j=1}^n{\mu}_{j t}{(PM)}_{j0}}\le {\varepsilon}_{4 kt}{\displaystyle \sum_{j=1}^n{\delta}_{j t f}^{*}}{(PM)}_{j0}\hfill & (60)\hfill \\ {}{\displaystyle \sum_{j=1}^n{\mu}_{j t}}{(FR)}_{j0}\ge \left\{\begin{array}{l}{\theta}_{4 kt}{(FR)}_{k0}\kern1em t=1\hfill \\ {}{\theta}_{4 kt}{\displaystyle \sum_{j=1}^n{\mu}_{j\left( t-1\right)}^{*}{(FR)}_{j0}\kern1em t>1}\hfill \end{array}\right.\hfill & (61)\hfill \\ {}{\displaystyle \sum_{j=1}^n{\mu}_{j t}}{(PCI)}_{j0}\le {(PCI)}_{k0}\hfill & (62)\hfill \\ {}{\displaystyle \sum_{j=1}^n{\mu}_{j t}}=1\hfill & (63)\hfill \\ {}{\varepsilon}_{4 kt}+{\theta}_{4 kt}=2\hfill & (64)\hfill \\ {}{\mu}_{j t}\ge 0\kern2.25em j=1,2,\dots, n\hfill & (65)\hfill \\ {}{\varepsilon}_{4 kt},{\theta}_{4 kt}\ge 0\hfill & (66)\hfill \end{array} $$

The objective function (Eq. 59) minimizes the relative efficiency of the fourth stage model of commuter rail system k during iteration t or equivalently maximizes its approximate inverse efficiency. The first constraint (Eq. 60) ensures that the hypothetical target commuter rail system for commuter rail system k during iteration t consumes no more of the intermediate product (PM) than is generated by the third stage during the forward pass of iteration t. The second constraint (Eq. 61) ensures that the hypothetical target commuter rail system for commuter rail system k during iteration t produces at least as much of the output (FR) as it does during iteration t − 1. The third constraint (Eq. 61) ensures that the hypothetical target commuter rail system for commuter rail system k during iteration t operates under conditions with no more of the site characteristic (PCI) than commuter rail system k.

We formulate the third stage VRS model for commuter rail system k during the backward pass of iteration t as follows:

$$ \begin{array}{ll}\mathrm{Min}\ {\varepsilon}_{3 ktb}\ \mathrm{or}\ \mathrm{Max}\ {\theta}_{3 ktb}\hfill & (67)\hfill \\ {}\mathrm{subject}\ \mathrm{to}\hfill & \hfill \\ {}{\displaystyle \sum_{j=1}^n{\delta}_{j t b}}{(RM)}_{j0}\le {\varepsilon}_{3 ktb}{\displaystyle \sum_{j=1}^n{\delta}_{j t f}^{*}}{(RM)}_{j0}\hfill & (68)\hfill \\ {}{\displaystyle \sum_{j=1}^n{\delta}_{j t b}}{(PM)}_{j0}\ge {\theta}_{3 ktb}{\displaystyle \sum_{j=1}^n{\mu}_{j t}^{*}}{(PM)}_{j0}\hfill & (69)\hfill \\ {}{\displaystyle \sum_{j=1}^n{\delta}_{j t b}}{(PD)}_{j0}\le {(PD)}_{k0}\hfill & (70)\hfill \\ {}{\displaystyle \sum_{j=1}^n{\delta}_{j t b}}{(ST)}_{j0}\le {(ST)}_{k0}\hfill & (71)\hfill \\ {}{\displaystyle \sum_{j=1}^n{\delta}_{j t b}}=1\hfill & (72)\hfill \\ {}{\varepsilon}_{3 ktb}+{\theta}_{3 ktb}=2\hfill & (73)\hfill \\ {}{\delta}_{j t b}\ge 0\kern2.75em j=1,2,\dots, n\hfill & (74)\hfill \\ {}{\varepsilon}_{3 ktb},{\theta}_{3 ktb}\ge 0\hfill & (75)\hfill \end{array} $$

The objective function (Eq. 67) minimizes the relative efficiency of the third stage model of commuter rail system k during the backward pass of iteration t or equivalently maximizes its approximate inverse efficiency. The first constraint (Eq. 68) ensures that the hypothetical target commuter rail system for commuter rail system k during the backward pass of iteration t consumes no more of the intermediate product (RM) than is consumed by the third stage during the forward pass of iteration t. The second constraint (Eq. 69) ensures that the hypothetical target commuter rail system for commuter rail system k during the backward pass of iteration t produces at least as much of the intermediate product (PM) as is consumed by the fourth stage operation during iteration t. The third and fourth constraints (Eqs. 70 and 71) ensure that the hypothetical target commuter rail system for commuter rail system k during the backward pass of iteration t operates under conditions with no more of the site characteristics (PD and ST) than commuter rail system k.

We formulate the second stage VRS model for commuter rail system k during the backward pass of iteration t as follows:

$$ \begin{array}{ll}\mathrm{Min}{\varepsilon}_{2 ktb}\ \mathrm{or}\ \mathrm{Max}\ {\theta}_{2 ktb}\hfill & (76)\hfill \\ {}\mathrm{subject}\ \mathrm{to}\hfill & \hfill \\ {}{\displaystyle \sum_{j=1}^n{\omega}_{j tb}}{(VM)}_{j0}\le {\varepsilon}_{2 ktb}{\displaystyle \sum_{j=1}^n{\omega}_{j tf}^{*}}{(VM)}_{j0}\hfill & (77)\hfill \\ {}{\displaystyle \sum_{j=1}^n{\omega}_{j tb}}{(RM)}_{j0}\ge {\theta}_{2 ktb}{\displaystyle \sum_{j=1}^n{\delta}_{j tb}^{*}}{(RM)}_{j0}\hfill & (78)\hfill \\ {}{\displaystyle \sum_{j=1}^n{\omega}_{j tb}}{(VA)}_{j0}\le {(VA)}_{k0}\hfill & (79)\hfill \\ {}{\displaystyle \sum_{j=1}^n{\omega}_{j tb}}{(DRM)}_{j0}\le {(DRM)}_{k0}\hfill & (80)\hfill \\ {}{\displaystyle \sum_{j=1}^n{\omega}_{j tb}}=1\hfill & (81)\hfill \\ {}{\varepsilon}_{2 ktb}+{\theta}_{2 ktb}=2\hfill & (82)\hfill \\ {}{\omega}_{j tb}\ge 0\kern2.25em j=1,2,\dots, n\hfill & (83)\hfill \\ {}{\varepsilon}_{2 ktb},{\theta}_{2 ktb}\ge 0\hfill & (84)\hfill \end{array} $$

The objective function (Eq. 76) minimizes the relative efficiency of the second stage model of commuter rail system k during the backward pass of iteration t or equivalently maximizes its approximate inverse efficiency. The first constraint (Eq. 77) ensures that the hypothetical target commuter rail system for commuter rail system k during the backward pass of iteration t consumes no more of the intermediate product (VM) than is consumed by the second stage during the forward pass of iteration t. The second constraint (Eq. 78) ensures that the hypothetical target commuter rail system for commuter rail system k during the backward pass of iteration t produces at least as much of the intermediate product (RM) as is consumed by the third stage during the backward pass of iteration t. The third and fourth constraints (Eqs. 79 and 80) ensure that the hypothetical target commuter rail system for commuter rail system k during the backward pass of iteration t operates under conditions with no more of the site characteristics (VA and DRM) than commuter rail system k _.

The above formulations describe one complete iteration for commuter rail system k. The process iterates until all parameters converge. We use Microsoft Excel©, the standard Solver© add-in, and Visual Basic for applications© programming language to construct the mathematical models and implement the iterative process to obtain the organizational efficiencies and optimal (target) values for all the parameters (OP^*, VM^*, RM^*, PM^*, and FR^*).