The Efficiency Measurement of Parallel Production Systems: A Non-radial Data Envelopment Analysis Model

Problem statement: Data Envelopment Analysis (DEA) is a non-parametric technique for measuring the relative efficiency of a set of production systems or Decision Making Units (DMU) that have multiple inputs and outputs. Sometimes, DMUs have a parallel structure, in which systems composed of parallel units work individually; the sum of their own inputs/outputs is the input/output of the system. For this type of system, conventional DEA models treat each DMU as a black box and ignore the performance of its units. Approach: This study introduces a DEA model in Slacks-Based Measure (SBM) formulation which considers the parallel relationship of the units within the system in measuring the efficiency of the system. Under this framework, the overall efficiency of the system is expressed as a weighted sum of the efficiencies of its units. Results: As an SBM model, the proposed model is non-radial and is suitable for measuring the efficiency when inputs and outputs may change non-proportionally. A theoretical result shows that if any unit of a parallel system is inefficient then the system is inefficient. Conclusion: This study introduces a non-radial DEA model, takes into account the operation of individual components within the parallel production system, to measure the overall efficiency as well as the efficiencies of sub-processes. This helps the decision makers recognize inefficient units and make later improvements.


INTRODUCTION
Data Envelopment Analysis (DEA) is a linear programming methodology in Operations Research and Economics that is extensively applied by various research communities (Sohn and Moon, 2004;Seol et al., 2007;Saljooghi, 2010 Zreika andElkanj, 2011). The domain of inquiry of the DEA is a set of production systems or decision making units (DMU), which use multiple inputs to produce multiple outputs. The aim of the DEA is to measure the relative efficiency of each DMU within a data set. The results specify how efficient each DMU has performed as compared to other DMUs in converting inputs to outputs. An issue which is of greater concern to the inefficient DMUs is what factors cause the inefficiency. Several studies have assigned to breaking down the overall efficiency into components so that the sources of inefficiency can be identified. One type of decomposition emphasizes the sub-processes of the production process. In recent years, a great number of DEA studies have focused on two-stage production systems, where all outputs from the first stage are intermediate products that make up the inputs to the second stage. For example see (Sexton and Lewis, 2003;Chen and Zhu, 2004;Liang et al., 2006;Kao and Hwang, 2008;Chen et al., 2009) among others. Recently, Tone and Tsutsui (2009) and Cook et al., (2010) have proposed DEA models for measuring the efficiency of network systems connected in series with linking activities.
In some situations, DMUs have a parallel structure that is composed of a set of units that work individually; the sum of their own inputs/outputs is the input/output of the DMUs. A typical example of these production systems is a university with faculties. The overall efficiency of the university can be calculated by the total inputs used and total outputs produced by all faculties. Each specific faculty can have an efficiency measured by comparing it with the equivalent faculties of other universities. The study of Färe and Primont (1984), which discusses the efficiency of firms with multiple plants, is probably the first study of such DMUs. Kao (1998) applied the Färe and Primont's methodology for measuring the efficiency of forest districts with multiple working circles in Taiwan. Castelli et al. (2004) discussed a hierarchical structure in which if there is only one layer, it becomes a parallel system. The works of Färe et al. (1997), Tsai and Molinero (2002) and Yu (2008) extend of the independent parallel system where certain resources are shared by some units.
Conventional DEA models view this type of production system as a black box and ignore the operations of its units. Recently, Kao (2009) has modified a standard DEA model and introduced a radial DEA model that evaluates the overall efficiency of the system as well as the efficiencies of its units. His method decomposes the inefficiency slack of a DMU into the inefficiency slacks of its sub-DMUs. As an application of Kao's parallel model,  examine the performance of the universities in Iran via a parallel production process.
This study presents an alternative method for estimating the efficiency of a parallel production system and the efficiency of its units. Since DEA models implicitly use Production Possibility Set (PPS) to evaluate the efficiency of DMUs, we first define the PPS of the parallel production systems. Then, based on this PPS, we introduce a non-radial DEA model in Slacks-Based Measure (SBM) formulation for aggregating the units in a parallel production system. Under this framework, the overall efficiency of the system is expressed as a weighted sum of the efficiencies of its units. With decomposition of the overall efficiency, the units which cause the inefficient operation of the system can be identified for future improvements An example from the forest production industry in Taiwan is applied to compare the new approach with Kao's parallel model.

MATERIALS AND METHODS
Production possibility set: Suppose we have n DMUs, where each DMU j (j=1,…,n) uses m inputs x ij (i = 1, …, m) to produce s outputs y rj (r = 1,…, s). It is assumed that all inputs and outputs are positive. We denote the DMU j by (x j , y j ), where x j = (x 1j , x 2j ,…, x mj ) T and y j = (y 1j , y 2j ,…, y mj ) T are input and output vectors, respectively. The Production Possibility Set (PPS) T is defined as a set of all inputs and outputs of a production technology in which outputs can be produced from inputs. Under the assumption of Constant Returns to Scale (CRS) the PPS can be represented as follows: is the intensity vector. The PPS under Variable Returns to Scale (VRS) assumption can be defined by adding the convexity constraint n j j 1 Definition 1: (Dominance). We say that DMU p (x p , y p ) dominates DMU q (x q , y q ) if and only if x p ≤x q and y p ≥y q with strict inequality holding for at least one component in the input or the output vector.
Thus, a DMU of the PPS is not dominated if and only if there is no other DMU (original or virtual) in the PPS which satisfies Definition 1. Radial and non radial DEA model: DEA provides for two types of measure: radial and non-radial. Radial models assume proportional change of inputs or outputs and usually disregard the existence of slacks in measuring efficiency scores. Radial measures are represented by CCR (Charnes et al., 1978) and BCC (Banker et al., 1984) Non-radial models, on the other hand, regard the slacks of each input or output and the variations of inputs and outputs as not proportional; in other words in non-radial models the inputs/outputs are allowed to decrease/increase at different rates. Non-radial models include Russell measure (Färe and Lovell, 1978) and Slacks-Based Measure (SBM) (Tone, 2001).
For evaluating the efficiency of DMU o (o ∈ {1,…,n}) the input-oriented CCR model, whose purpose is to minimize input custom while keeping the level of current outputs, is set as follows: The optimal solution of ρ* is the SBM efficiency score. It can be obviously identified that 0<ρ*≤1 and supports the properties of unit invariance and monotone.
Suppose an optimal solution for model (3)  The input (output)-oriented SBM model can be defined by ignoring the denominator (numerator) of the objective function.
Here we emphasize that, as demonstrated by Tone (2001), the efficiency score measured by the SBM model is not greater than the efficiency score measured by the CCR model. Moreover, a DMU is SBM-efficient if and only if it is CCR-efficient.
Parallel production system: Consider a parallel production process as shown in Fig. 1. Suppose we have n DMUs, of which each DMU j (j=1,…,n) is composed of t j units (sub-DMUs) connected in parallel. Each sub-DMU uses the same inputs to produce the same outputs, individually. Sub-DMU p Kao's parallel model: Kao (2009) has developed a DEA model based on the input-oriented CCR multiplier model such that by minimizing the inefficiency slack instead of maximizing the efficiency, we are able to decompose the inefficiency slack of a DMU into the inefficiency slacks of its sub-DMUs. Kao's model for measuring the inefficiency of DMU o is given by: where, (*) shows the optimal value from model (5).

The parallel SBM model:
The PPS parallel C T under the CRS assumption for n×t j sub-DMUs of n parallel production systems, is defined by: j j t t n n parallel p p p p p C j j j j j j 1 p 1 j 1 p 1 T ( x , y ) x λ x , y λ y , λ 0

∑∑ ∑∑
Note that if all Suppose, DMU o (o∈{1,….,n}) to be the DMU under evaluation. In an effort to measure the overall efficiency of DMU o , first by using the input-oriented SBM model, we calculate the efficiency score of each sub-DMU p (p=1,…,t o ) as follows: After evaluating the efficiency scores of all Sub-DMUs, we define the overall efficiency score of DMU o as a weighted combination of the efficiency scores of its sub-DMUs. This is shown in the following equation: where, the weight Definition 6: DMU o is said to be efficient if its overall efficiency score is equal to one, i.e., o E 1 = .
Decomposing the overall efficiency of a system into the weighted combination of its unit's efficiencies helps us to identify the units that cause inefficiency. By using the model (7), we are able to recognize the inefficient sub-DMUs and make later improvements. Also, using Eq. 8 we can evaluate the overall efficiency of the DMU o in a way that takes into account the operations of all its sub-DMUs.
Note that, similarly, we can introduce an outputoriented SBM model for parallel production systems such that the weights associated to sup-DMU of DMU o can be defined as:

RESULTS
The following theorem explains the relationship between the efficiency of a parallel production system and its production units.
As a contraposition to Theorem 1, we have.
Empirical example: Now, we apply the proposed model to the national forests of Taiwan as studied by Kao (2009). In Taiwan, the forest lands are divided into eight regions, each of which is divided into four or five sub-regions called working circles (WCs). These WCs are the basic component in the management of the forest. The forest production process is a characteristic parallel production process, in that each region has several subordinated WCs operating individually. There are four inputs: The data are shown in Table 1. For each input (output) the quantity of a region is the sum of its sub-regions.
The results of this measurement of efficiency are reported in Table 2, where the second column shows the weights calculated from (9) for each sub-DMU, the third column is the efficiency score calculated using model (7) and Eq. 8. The efficiency scores of the eight DMUs calculated by the input-oriented of model (3) without taking into account the operations of sub-DMUs are shown in the last column under the heading conventional SBM model.

DISCUSSION
As pointed out by many authors including Kao and Hwang (2008), Kao (2009), Chen et al. (2009), Tone and Tsutsui (2009) and Cook et al. (2010, the conventional DEA models apply a single process to evaluate the transforming efficiency of multiple inputs and outputs such that they fail to measure the efforts of different processes and sub-processes within the production systems. Thus, we cannot evaluate the impact of sub-process inefficiencies on the overall efficiency of the system as a whole. In these cases, it is possible that the conventional DEA models evaluate a system as efficient even if none of its component processes is efficient From Table 2, it can be seen that six DMUs are efficient under the conventional SBM model while according to Theorem 1, under the parallel SBM model, since none of DMUs performs efficiently in all its own sub-DMUs, none of them performs efficiently as a whole. Thus, by using the results of this efficiency measurement we are able to identify the inefficient sub-DMUs and make future improvement. The rankings of the overall efficiency scores of the eight regions taking our approach and taking Kao's approach are shown in Table 3. Comparison of the two sets of scores shows then to have almost identical ranking. The Spearman Rank Correlation coefficient for the rankings in Table 3 is 0.976, showing that the correlation between our results and Kao's results is very high.    Thus the new approach is suitable for measuring the overall efficiency of the whole system with the added benefit of allowing inefficient sub-DMUs to be identified and potentially rectified.

CONCLUSION
In an earlier study, a radial DEA model was introduced by Kao (2009) for measuring the efficiency of a system composed of parallel units operating independently and where the sum of inputs/outputs for all units is equal to the input/output of the system. In this study, we have introduced a non-radial model based on a Slacks-Based Measure (SBM) framework that evaluates the overall efficiency of the system by considering the operations of its units. Under this framework, the overall efficiency of the system is expressed as a weighted sum of the efficiencies of its units. With decomposition of the overall efficiency, the units which cause the inefficient operation of the system can be identified for future improvements The proposed model is based on the assumption of Constant Returns to Scale (CRS). By adding the convexity constraint into the PPS which is built by n×t j sub-DMUs, the discussion can be expanded to use the Variable Returns to Scale (VRS) assumption.
It is noteworthy that real systems are generally more complex than the parallel system discussed in this study. Tone and Tsutsui (2009) developed a network DEA model based on a weighted SBM approach that can be applied in series systems. Since the series and parallel structure are two basic structures of a network system, we can transform a network system into a combination of series and parallel structures to evaluate the overall efficiency and the efficiencies of sub-processes.