Dominance network analysis of economic efficiency

Highlights

• • A dominance network can be built based on the observed input and output data.

• • Two types of dominance relationships are considered: technical and economic.

• • Complex networks analysis can be performed on this layered, weighted, directed network.

• • Technical, profit and allocative inefficiencies can be computed.

• • Profit inefficiency relation has an underlying scalar potential.

Abstract

This paper proposes an enhanced Dominance Network (DN) to assess the technical, economic and allocative efficiency of a set of Decision Making Units (DMUs). In a DN, the nodes represent DMUs and the arcs correspond to dominance relationships between them. Two types of dominance relationship are considered: technical and economic. The length of a technical dominance arc between two nodes is a weighted measure of the input and output differences between the two DMUs. The length of an economic dominance arc between two nodes corresponds to the cost, revenue or profit difference between them (depending on whether only unit input prices, unit output prices or both are known). The proposed dominance network is a multiplex network with two relations; the structure of both relations is similar. Thus, both of them are layered and their arcs have transitivity and additivity properties. However, since technical dominance implies economic dominance but not the reverse, economic dominance is more common and has a deeper structure. It may also have an underlying potential field so that the length of the arcs between any two nodes depends on the difference in their potentials and the direction of the arcs depends on the sign of that difference. Allocative inefficiencies can also be gauged on this DN. Complex network measures can be used to characterize and study this type of DN.

Introduction

Non-parametric Frontier Analysis methods (of which Data Envelopment Analysis, DEA, is probably the best known) assess the relative efficiency of DMUs. These methods see the DMUs as production processes that transform inputs into outputs. Moreover, they are data-driven so that only the data on the input consumption and output production of each DMU are required. DEA approaches have been extensively studied and applied as can be seen in the existing textbooks on the subject (e.g. Cooper et al., 2004, Cooper et al., 2006, Färe et al., 1985, Färe et al., 1994).

Recently, Calzada-Infante and Lozano (2016) used DN to assess the medal winning performance of nations at Olympic Games. The corresponding DN is characterized using different complex network measures, such as node strength, clustering coefficient, betweenness centrality, degree-degree correlation, etc. This paper proposes an enhanced DN analysis that includes not only technical efficiency but also economic (cost, revenue or profit) efficiency. Technical efficiency (TE) basically measures the maximum input reduction and output increase that a given DMU may achieve. Economic efficiency, on the other hand, refers to the maximum cost reduction or revenue or profit increase that a given DMU may achieve. TE assessment only requires input and output data while economic efficiency assessment also requires knowing the outputs and inputs unit prices.

Since TE assessment does not use price information, the projection of a DMU is done considering only the operating points that dominate it in technical terms, i.e. in terms of the input consumption and output production. This leads to the technical dominance criterion. Economic efficiency, on the other hand, makes use of price information, allowing certain inputs to be increased or certain outputs to be decreased if that leads to an overall cost, revenue or profit improvement, depending on whether we are considering cost, revenue or profit efficiency. In any case, economic efficiency leads to an economic dominance criterion which is different from that of technical dominance and it is based on the corresponding cost, revenue or profit. Hence, in the proposed approach we consider a DN with two different dominance relations, one based on TE and the other based on economic efficiency.

The proposed enhanced DN approach is a novel way of assessing the economic performance of DMUs using complex networks tools. Some researchers have previously applied complex network analysis in a DEA context but their approach was not based on the DN concept. Thus, for example, complex network analysis has been proposed to rank efficient DMUs (Ho, Liu, Lu, & Huang, 2014; Leem & Chun, 2015; Liu & Lu, 2010; Liu et al., 2009, Liu et al., 2014). In these approaches, the network considered is weighted and directed and it is constructed based on the optimal intensity variables (commonly known as lambda parameters) computed using an envelopment DEA model, sometimes with different input-output specifications. The nodes of this network are the DMUs and the arc weights are the optimal values of the intensity variables. If only one input-output specification is used, the resulting network is bipartite, with the arcs going from each inefficient DMU to the efficient DMUs in its peer group. Two different centrality measures are proposed to rank the efficient DMUs.

Ghahraman and Prior (2015) use a different approach aimed at selecting an optimal stepwise benchmark path from an inefficient operating unit (OU) to the efficient frontier. Their network considers that the nodes are the DMUs but this time an arc between two nodes r and j exists if, and only if, OU j has a higher efficiency score than OU r. The corresponding arc weight is computed using a weighted measure that takes into account the input similarities between the two DMUs, their efficiency gap (modified using an exponential penalty function) and a fixed cost for each link. They compute shortest paths on this network as well as clusters of DMUs (based on the maximum input and output percentage changes in a single step). They also use complex network analysis to discriminate between efficient and intermediate DMUs, clustering the DMUs and identifying possible outliers and specialized units.

The DN approach first used in Calzada-Infante and Lozano (2016) is different from those commented on above as it constructs a DN based on the dominance relationships between the DMUs. Thus, an arc between a node r and a nodejexists if, and only if, DMUj(weakly) dominates DMU r, i.e. if DMUjconsumes less (or at most the same amount of) inputs and produces more (or at least an equal amount of) outputs than DMU r. Calzada-Infante and Lozano (2016) use what can be called a Normalized Additive Inefficiency (NAI) metric to compute the arc weights and implicitly assumes a basic Free Disposal Hull (FDH) technology. However, other DEA metrics and other DEA technologies can also be considered.

In this paper, an enhanced DN is considered for the cases in which, in addition to the input and output data, unit input prices, unit output prices or both, are known. In those cases, in addition to the technical dominance relationships, economic dominance relationships can be defined. Thus, the relationships between the DMUs are analysed from two different points of view: technical and economic. The resulting DN integrates both relations and its analysis, using complex network tools, provides an innovative approach to economic efficiency assessment. Thus, while in DEA the TE score measures the distance to the frontier, i.e. the difference between a DMU and its efficient benchmark, the DN approach also considers the existence of links between any two inefficient DMUs if one dominates the other in technical terms. The length of the corresponding arc measures the relative inefficiency between them, i.e. the difference in their respective TE. Similarly, while in DEA the economic efficiency of a DMU refers to the maximum cost, revenue or profit improvement for a given DMU, in the enhanced DN approach the lengths of the arcs of the economic efficiency relation correspond to the difference in cost, revenue or profit between the origin and the destination nodes. By following a directed path along the DN, successive improvements in TE or in economic efficiency (depending on the dominance relation considered) can be obtained. Thus, the proposed approach allows a graphical and quantitative representation of the TE and economic efficiency of the DMUs in the sample and of the possible improvement paths that can be followed.

This double graphical plus quantitative feature of the proposed DN approach is useful because most DEA problems involve multiple inputs and outputs observations whose direct visualization in a multidimensional space is not possible. Particularly interesting is the DN approach in those cases in which the DMUs belong to the same organization (e.g. bank branches, retail stores, bus routes, etc) because this analysis tool allows a global perspective of the problem, i.e. a systemic view of the dataset, at the same time that the relative performance and relative position of each individual DMU within the whole is ascertained. Another situation in which this tool may be useful is in a competing DMU scenario (e.g. airlines, mutual funds, etc) in which the visualization capability of DN allows a sort of efficiency positioning map of the different DMUs so that a strategic analysis of the technical and economic efficiency status and relative position of each DMU may be assessed.

The structure of this paper is the following. In Section 2 a review of the literature is carried out. In Section 3 the proposed approach is presented and discussed. Section 4 illustrates the proposed approach on a simple dataset while Section 5 presents a real world application in the banking sector. Section 6 summarizes and concludes.