A Non-Radial Input Relaxation Model Approach for Evaluating Input Congestion in Data Envelopment Analysis

Introduction And Review Of The Literature

Data envelopment analysis (DEA) is a nonparametric approach to evaluate the relative efficiency of decision making units (DMUs) which was originated by Charnes et al., 1978. The first model in DEA was referred to as the CCR model. Since 1978, further models have been introduced in the literature, for instance, Banker et al. (1984) extended a variable returns to scale version of the CCR model, known as the BCC model in the literature. Charnes et al. (1985) also introduced additive models in DEA. Furthermore, Khodabakhshi, Gholami & Kheirollahi. (2009) developed an additive model to provide an alternative approach for estimating returns to scale (RTS) in DEA. Their model is developed in both stochastic and fuzzy DEA.

Khodabakhshi. (2011) has proposed an input relaxation approach to rank efficient DMUs in stochastic DEA. Additional papers on stochastic and fuzzy DEA, can be found in Kheirollahi et al, 2015; Khodabakhshi, 2010; and Eslami, 2013. The performances of DMUs are affected by the level of sources they utilize. For example, congesting sources will likely inhibit the production of outputs beyond those normally observed in the DMU being evaluated. It is important to identify DMUs that have input congestion as well as to estimate the actual value (s) of their input congestion. Congestion is a particularly severe form of inefficiency in DMUs, refers to situations where reductions in one or more inputs generate an increase in one or more outputs.

The Quran, as the holy book for Muslim people, has emphasized, on economic issues in particular, the prevention of increment costs and input congestion in several verses. For example, this holy book suggest that the people, children of Adam, eat and drink, yet do not overdo things; Allah doesn’t approve of the extravagant. See the chapter of ‘Heights’ in the Quran, verse 31, Irving, 2005.

Fare and Svensson (1980) have extended the topic of congestion as it relates to the law of variable proportions. Fare & Grosskopf (1983) subsequently extended and developed this in the context of en DEA, which gives the Fare-Svensson model an operationally implementable form. Later, Fare, Grosskopf & Lovell (1985) discussed DEA-related models and methods (viz., the FGL approach) for production efficiency evaluation. Cooper et al, (2000) introduced a unified additive model approach for evaluating inefficiency and congestion with associated measures in DEA. Kheirollahi et al. (2015) used a BCC input relaxation model to identifying the input congestion of Iranian hospitals in stochastic DEA with selected chance constrained programming approaches.

Cooper et al, (2001b) have extended previous approaches to congestion in order to identify managerial inefficiency as a possible additional source of output shortfalls in the presence of congestion. This was done in a manner that made it possible to reduce some of the congesting inputs while increasing (or, at least not decreasing) other inputs that are also congesting. This research, following Patrick et al, (2004) distinguishes between congestion and other types of inefficiency. Tone & Biresh (2004) make a novel attempt to suggest a method within a nonparametric framework to measure scale elasticity in production, but in the presence of congestion. Sueyoshi and Sekitani (2008) explore how to deal with the occurrence of multiple solutions in the DEA-based congestion measurement.

The current study proposes a new approach for congestion measurement, and theoretically compares the proposed approach with that of Tone and Biresh (2004). Noura et al. (2010) focus on the latter work in proposing a new method that requires considerably less computation. Then, by proving a selected theorem, it shows that the proposed methodology is indeed equivalent to that of Cooper et al. (2002). Quanling and Hong (2011) have introduced the concept of return to scale (RTS) as a typically important concept in production input and output analyses. It measures the marginal returns of additional input of a DMU or production unit.