Computing multiperiod efficiency using dominance networks

Abstract

Dominance Network (DN) analysis has been proposed to assess the efficiency of a set of operating units (OUs) that consume inputs to produce outputs. In a DN, the nodes represent OUs and the arcs correspond to dominance relationships (DR) between them. So far, this analysis has been applied in static scenarios where the OUs belong to the same time period. In this paper, the methodology is extended to the case in which the input and output data refer to multiple time periods tracking the performance evolution of each OU. The proposed Temporal Dominance Network (TDN) contains a node for each OU in each time period. TDN is a weighted directed acyclic graph. There exist contemporaneous DR (between nodes that correspond to the same time period) as well as non-contemporaneous DR (between nodes that correspond to different time periods). The union of these two types of DR defines the TDN. All these DR are transitive, which allows us to represent the TDN by its corresponding skeleton. Various filters and node layouts that can be used to help visualise the TDN are proposed. In addition, a number of network metrics are used to characterise the efficiency of the various OUs in the different periods as well as the corresponding productivity change between any two periods.

Appendix

1.1 Notation

Let

$D = \left\{ {1,2, \ldots ,n} \right\}$:

: Set of OUs.

j,r,p = 1,2,…,n: :

Indexes on OUs.

i = 1,2,…,m: :

Index on input dimensions.

k = 1,2,…,s: :

Index on output dimensions.

$x_{j} = \left( {x_{1j} ,x_{2j} , \ldots x_{mj} } \right)$:

: Input vector of OU j.

$y_{j} = \left( {y_{1j} ,y_{2j} , \ldots y_{sj} } \right)$:

: Output vector of OU j.

$D(r) = \{ j:\;x_{ij} \le x_{ir} \;\forall i\;\; \wedge \;y_{kj} \ge y_{kr} \;\forall k\;\; \wedge \;\left( {x_{j} ,y_{j} } \right) \ne \left( {x_{r} ,y_{r} } \right)\}$:

: Set of OUs that weakly dominate OU r.

$D_{{}}^{ - 1} (r) = \left\{ {f:r \in D\left( f \right)} \right\}$:

: Set of OUs that are dominated by OU r.

$D^{*} = \left\{ {r:D\left( r \right) = \emptyset } \right\}$:

: Set of non-dominated (i.e. efficient) OUs.

$D_{{}}^{*} (r) = \left\{ {\begin{array}{*{20}c} {\left\{ r \right\}} & {if\;r \in D^{*} } \\ {D_{{}} (r) \cap D^{*} } & {if\;r \notin D^{*} } \\ \end{array} } \right\}$:

: Set of efficient benchmarks of OU r.

$D_{{}}^{ - - 1} (r) = \{ f:f \in D^{ - 1} (r) \wedge D_{{}} (f) \cap D^{*} = \left\{ r \right\} \}$:

: Set of OUs dominated by r that have r as their only efficient benchmark.

1.2 Node-level measures

$d_{r}^{in} = \left| {D_{{}}^{ - 1} (r)} \right|$ :

: In-degree of node r.

$d_{r}^{out} = \left| {D(r)} \right|$ :

: Out-degree of node r.

$s_{r}^{in} = \sum\limits_{{f \in D_{{}}^{ - 1} (r)}}^{{}} {e_{fr} }$ :

: In-strength of node r.

$s_{r}^{out} = \sum\limits_{j \in D(r)}^{{}} {e_{rj} }$ :

: Out-strength of node r.

$e_{r}^{\max } = \mathop {\max }\limits_{{j \in D_{{}}^{*} \left( r \right)}} e_{rj}$ :

: Distance of node r to its farthest efficient benchmark.

$\tau_{r}^{\min } = \mathop {\min }\limits_{{j \in D_{{}}^{*} \left( r \right)}} \;e_{rj}$ :

: Distance of node r to its closest efficient benchmark.

$\eta_{r} = d_{r}^{out} + d_{r}^{in}$ :

: Specificity of node r.

$\gamma_{r} = d_{r}^{out} \cdot d_{r}^{in}$ :

: Hub index of node r.

1.3 Node-level measures for efficient nodes

$\left| {D_{{}}^{ - 1} (j)} \right| = d_{j}^{in}$ :

: Benchmarking count of node$j \in D^{*}$

$\left| {D_{{}}^{ - - 1} (j)} \right|$ :

: Benchmarking necessity of node $j \in D^{*}$

$\kappa_{j} = \sum\limits_{{r \in D_{{}}^{ - 1} (j)}} {e_{rj} }$ :

: Benchmarking potential of node $j \in D^{*}$

$\sigma_{j} = \mathop {\max }\limits_{{r \in D_{{}}^{ - 1} (j)}} \;e_{rj}$ :

: inefficiency radius of node $j \in D^{*}$

$\varphi_{j} = \left| {\left\{ {r:D(r) = \left\{ j \right\}} \right\}} \right|$ :

: superefficiency index of node$j \in D^{*}$

.

1.4 Network-level measures

$d_{{}}^{aver} = \frac{1}{\left| D \right|}\sum\limits_{r \in D}^{{}} {d_{r}^{in} } = \frac{1}{\left| D \right|}\sum\limits_{r \in D}^{{}} {d_{r}^{out} }$ :

: Average degree.

$\rho = \frac{{d^{aver} }}{\left| D \right| - 1}$ :

: Network density.

$\Delta = \mathop {\max }\limits_{(r,j) \in E} \;e_{rj}$ :

: Network diameter.

$\hat{\pi } = \frac{{\left| {D_{{}}^{*} } \right|}}{\left| D \right|}$ :

: Percentage of efficient nodes.

$\theta^{aver} = \frac{1}{\left| D \right|}\sum\limits_{r \in D} {e_{r}^{\max } }$ :

: Average inefficiency score.