Selecting common projection direction in DEA directional distance function based on directional extensibility

Highlights

• We consider to select common projection direction in nonparametric directional distance function.

• A new concept of directional extensibility is defined.

• A trade-off approach is proposed for common projection direction selection.

• The optimal direction ensures the organizations to make improvements in all performance metrics.

• A case study of China’s logistics companies is proceeded.

Abstract

This paper develops a new approach to select a common projection direction for performance evaluation of decision-making units (DMUs) using the data envelopment analysis (DEA) directional distance function. First, we define the concept of directional extensibility of a specific projection direction with respect to a set of inefficient DMUs. The concept shows the projection direction’s ability in reducing inefficiencies, simultaneously, for all inefficient DMUs. Using this concept, we propose a model which identifies the common projection direction that has the maximum directional extensibility. However, this common projection direction may contain zero elements. To avoid this problem and to select the final common projection direction, an algorithm is developed to select the common projection direction which is resulted in a trade-off between the directional extensibility and the similarity among the elements of the common projection direction. Our developments contribute by formally defining a concept (i.e., the directional extensibility) which can well reflect a common projection direction’s ability in reducing the DMUs’ inefficiencies. Moreover, the selected final common projection direction has both good directional extensibility and no zero elements, which helps to guide the inefficient DMUs to develop in a balanced way because the direction suggests the DMUs to improve in all inputs and outputs. Finally, we demonstrate the usefulness of our proposed approach by using an empirical case of 18 logistics companies listed among China’s top 500 enterprises in 2018.

Introduction

As a useful tool of efficiency and productivity measurement, the directional distance function (DDF), which was introduced by Chambers et al., 1996, Chambers et al., 1998, has attracted considerable interest from academia and practitioners (Wang, Xian, Lee, Wei, & Huang, 2019). Different from the input/output distance function proposed by Shephard, 1953, Shephard, 1970, which considers inputs decreasing or outputs increasing, DDF can expand desirable outputs and reduce inputs/undesirable outputs simultaneously when it is used for the efficiency and productivity evaluation of units which consume multiple inputs to produce multiple outputs (Chambers et al., 1998, Chen et al., 2013, Wang et al., 2019). Another advantage of DDF is the flexibility in choosing a directional vector, either an exogenous or endogenous direction, in efficiency estimation. Accordingly, it has been extensively used in the fields of economic growth analysis, productivity analysis, eco-efficiency measurement, and shadow price estimation (Zhang and Choi, 2014, Wang et al., 2019). The empirical studies and theoretical models based on DDF can incorporate either parametric methodology or nonparametric methodology, and the current study particularly considers the projection direction selection for DDF in the framework of data envelopment analysis (DEA), a typical and representative nonparametric linear-programming method (Charnes et al., 1978, Banker et al., 1984, Li et al., 2018, Toloo and Mensah, 2019, Zhu et al., 2020).

In DEA DDF, the production efficient frontier is used as the benchmark for the decision-making units (DMUs) to do performance improvement. Each DMU achieves efficient by moving along any projection direction towards the production efficient frontier. However, there is no consistent criterion for choosing an optimal direction along which to measure the efficiency and productivity of the DMU under evaluation (Färe et al., 2017, Yang et al., 2018) in DEA DDF. Hence, selecting an appropriate direction (e.g., endogenous direction) has been one of the key issues in the research field of DEA DDF. The research about projection direction selection approaches in DEA DDF can be classified as exogenous approaches and endogenous approaches (Wang et al., 2019). The former mainly relates to approaches in which the projection directions are predetermined by decisionmakers, and the latter refers to approaches in which the projection directions are selected via an endogenous criterion.

Generally, the exogenous projection direction selection approaches can be classified into three categories. One category commonly takes the observed input/output values as the projection direction (Chambers et al., 1996, Oum et al., 2013, Hampf and Krüger, 2014b). Consequently, each DMU has its own specific projection direction which is obtained depending on its input/output values. Another category usually takes the unit value as projection direction (e.g., Färe et al., 2006, Halkos and Tzeremes, 2013), i.e., the direction in which all the elements are 1 or −1. However, DDF based on either input/output observation value direction or unit value direction may overestimate the efficiency of the evaluated DMU since it cannot guarantee to project the focal DMU on the efficient frontier (Wang et al., 2019). Additionally, the inefficiency scores measured along the unit value direction fails to be unit invariant. Therefore, the last category of studies proposes to use conditional directions, which are built incorporating various perspectives (e.g., shadow price estimation), to improve the previous two categories of exogenous directions. For example, to estimate the shadow price of pollutant, the annual plans which consider production/environment inefficiency are taken as directions (Lee and Zhou, 2015). To make comparisons between DMUs easier, the average inputs/outputs of all DMUs are utilized as directions (Dervaux et al., 2009, Hailu and Chambers, 2012, Simar et al., 2012). However, Wang et al. (2019) pointed out that the conditional directions need to find more support from viewpoint of economic meaning and theory (Wang et al., 2019).

Exogenous direction selection approaches usually lack theoretical basis or economic implications, or even worse cause subjectivity issues of direction selection and sensitive evaluation results (Wang et al., 2019, Yang et al., 2018), so many scholars suggested using endogenous direction selection approaches. The exogenous direction selection approaches can be classified into theoretically optimized direction selection approaches and market-oriented direction selection approaches (Wang et al., 2019). The former mainly focuses on the directions toward the closest target or furthest target, and the latter determines the directions which relate to cost minimization, profit maximization, or marginal profit maximization. From the theoretical optimization perspective, many scholars used the least distance (or, closest target) approach to determine the projection direction which can help the inefficient DMUs to project to its closest target on the efficient frontier (Briec, 1999, Frei and Harker, 1999), and this type of approach was improved by Briec and Leleu, 2003, Aparicio et al., 2007, Baek and Lee, 2009, Amirteimoori and Kordrostami, 2010, Lozano and Soltani, 2018, Petersen, 2018, and Zhu, Wu, Ji, and Li (2018). Contrary to the perspective of the closest projection target, the furthest distance approaches select a direction towards the furthest target to identify the largest improvement potential for the inefficient DMUs (e.g. Färe and Grosskopf, 2010, Färe et al., 2013, Hampf and Krüger, 2014a, Adler and Volta, 2016, Krüger, 2018). Moreover, by incorporating the enterprise behaviors and price information, various approaches have been introduced to select projection direction from market-oriented perspectives. For example, the cost minimization direction selection approaches can help to select the projection direction so that an organization can improve to the frontier production which has the minimum cost (Ray and Mukherjee, 2000, Granderson and Prior, 2013). Additionally, the direction towards profit maximization provides a long-term development target for the enterprises (Zofio, Pastor, & Aparicio, 2013), while the direction towards marginal profit maximization gives a short-term target.

The endogenous direction selection approaches well complement the direction selection techniques in DEA DDF-based modeling and application by encompassing theoretically optimization and economics meanings. However, there are still several limitations occurring in endogenous approaches which were pointed out by Wang et al. (2019). For example, the theoretically optimized direction is hard to obtain in real production processes, and the prices of inputs/outputs in cost minimization/marginal profit maximization directions are basically exogenous. More importantly, the endogenous approaches generally require the DMUs to use their own projection direction when making frontier projection instead of using a common projection direction. This may lead to the problem that the DMUs may doubt the fairness of the evaluation result thus to reduce their motivation to accept the evaluation result, because a DMU with relatively poor performance on some of its input and output indicators may take the advantage of using its most favorable projection direction to generate an overestimated efficiency that results in itself being a dominator in the evaluation result (Kao and Hung, 2005, Wang et al., 2009, Wu et al., 2016, Yang et al., 2018). Besides, the balanced improvement of all input/output indicators is important and necessary to the sustainable development of inefficient DMUs (Wei, Chu, Song, & Yang, 2019).

To fill the research gaps discussed above, this paper develops a new approach to select a common direction for inefficient DMUs to do frontier projection based on a new concept of directional extensibility. The motivation of using the concept of directional extensibility is because it clearly defines a common projection direction’s ability to reduce all the inefficient DMUs’ inefficiencies. Then, we propose a model which can identify the common projection direction that has the maximum directional extensibility. However, the common projection direction with the maximum directional extensibility might contain zeros. To address this problem and to have a final common projection direction that has relatively good directional extensibility, we propose an algorithm which makes a trade-off between the goals of maximizing the common projection direction’s directional extensibility and maximizing the similarity among the elements in the common projection direction. The algorithm always generates a common projection direction that has considerable good directional extensibility and contains no zero elements. Containing no zero elements of the common projection direction guides the inefficient DMUs to do a balanced improvement when improving efficiencies since all the input and output indicators need to be improved.

The rest of the paper is structured as follows. Section 2 gives the preliminaries which contain brief introductions of the directional distance function and the least-distance model. Section 3 proposes the new methodology for common projection direction selection and a new model for efficiency evaluation of the DMUs utilizing the selected common projection direction. Further, in Section 4, we apply our proposed approach to a case study of 18 Chinese logistics companies to illustrate its merits. Finally, conclusions and several further research directions are provided in Section 5.