Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 247248, 9 pages

http://dx.doi.org/10.1155/2014/247248

## Dimension-Specific Efficiency Measurement Using Data Envelopment Analysis

Institute of Command and Information Systems, PLA University of Science and Technology, Nanjing 210007, China

Received 26 October 2014; Revised 6 December 2014; Accepted 6 December 2014; Published 25 December 2014

Academic Editor: Wei-Chiang Hong

Copyright © 2014 Hongjun Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Data envelopment analysis (DEA) is a powerful tool for evaluating and improving the performance of a set of decision-making units (DMUs). Empirically, there are usually many DMUs exhibiting “efficient” status in multi-input multioutput situations. However, it is not appropriate to assert that all efficient DMUs have equivalent performances. Actually, a DMU can be evaluated to be efficient as long as it performs best in a single dimension. This paper argues that an efficient DMU of a particular input-output proportion has its own specialty and may also perform poorly in some dimensions. Two DEA-based approaches are proposed to measure the dimension-specific efficiency of DMUs. One is measuring efficiency in multiplier-form by further processing the original multiplier DEA model. The other is calculating efficiency in envelopment-form by comparing with an ideal DMU. The proposed approaches are applied to 26 supermarkets in the city of Nanjing, China, which have provided new insights on efficiency for the managers.

#### 1. Introduction

Data envelopment analysis (DEA) [1] is a mathematical programming method for evaluating the relative efficiency of decision-making units (DMUs) with multiple outputs and multiple inputs. By using DEA, a single index (namely, efficiency score) can be obtained from the ratio of weighted outputs to weighted inputs, as an assessment of a DMU’s overall performance.

Introduced by Charnes et al. in 1978 [1], DEA has been recognized as an excellent and easily used methodology for modeling operational processes for performance evaluations [2]. It has been widely applied in many application areas [3], such as banks [4, 5], agriculture [6, 7], health care [8, 9], education [10, 11], and transportation [12, 13].

There are several reasons why DEA can be successfully applied [3, 14]. First, it is a nonparametric technique that does not require any underlying assumptions on the production function defining the relationship between inputs and outputs. Second, it can distinguish between efficient and inefficient DMUs. Third, multiple inputs and outputs can be considered simultaneously. Finally, it can pinpoint the sources and the amount of deficiency for each of the inefficient DMUs and provide proper benchmarking information for them to make improvement.

However, in the extensive applications, most attentions have been paid on the evaluation of overall performances, whereas no literature focuses on how a DMU performs in an individual input/output dimension. Since a single index is used to evaluate a DMU’s performance, efficient DMUs, with a maximum efficiency score of unity, are usually considered perfect with no room for improvement. Actually, it is not always the case. Empirically, there are usually multiple units exhibiting “efficient” status in multi-input and multioutput situations [15]. In this study, we argue that efficient DMUs of different input-output proportions have different production specialties and may also perform poorly in some aspects. With the flexibility of weights selection, a DMU can achieve an “efficient” status as long as it performs best in very few dimensions, no matter how poor it is in other dimensions. A larger weight can be assigned to its best-practice dimensions and an infinitesimal value (even zero, if allowed) to its weak points.

Furthermore, as we all know, DEA is sensitive to variable selection [16]. Measuring efficiency in each dimension, to some extent, can reduce the sensitivity.

Therefore, it is of great significance to measure dimension-specific efficiency of DMUs. A simple example can be used to illustrate this problem. Suppose that there are two supermarkets A and B in the same metropolitan area, each using two inputs (promotion expenses and number of employees) to produce the same amount of outputs. A spends less in promotion but has more employees than B. Then both A and B will be evaluated as efficient by the original DEA model. After investigation we find that A is good at promotion, whereas B is good at staff management. Therefore, more information can be obtained by measuring efficiency in each dimension.

This paper proposes two approaches for measuring the dimension-specific efficiency of DMUs. One is to measure efficiency in a multiplier-form by further processing the original multiplier DEA model. The other is to calculate efficiency in an envelopment-form by comparing with an ideal DMU. The two approaches evolve from the “multiplier side” and the “envelopment side” of the original DEA model, respectively. Different ideas are used for the definition of dimension-specific efficiency, so the new models are not dual problems. Therefore, efficiency scores calculated by the two approaches will not be necessarily the same.

The remainder of the paper is organized as follows. Sections 2 and 3 present the two proposed approaches, respectively. Both approaches are demonstrated by the same simple numerical example. Section 4 applies the two approaches to a real world data set consisting of 26 supermarkets. Finally, Section 5 summarizes our work.

#### 2. Multiplier Models for Dimension-Specific Efficiency Measurement: Approach 1

##### 2.1. Multiplier CCR Model

Assume there are DMUs under evaluation. Each () consumes inputs () to produce outputs (). The efficiency score for any evaluated is defined as the maximal ratio of its total weighted output to its total weighted input, under the constraints that the efficiency for each DMU cannot exceed unity. The mathematical programming problem can thus be stated as the following multiplier CCR model [1]: where and are multipliers to the outputs and inputs, respectively.

This fractional programming problem can be transformed into a linear programming problem using the Charnes-Cooper transformation [1] as Model (2) is input-oriented and under constant returns-to-scale (CRS) assumption. The performance of is usually considered to be CCR DEA efficient if and CCR DEA inefficient if .

##### 2.2. Evaluation

A characteristic of the original DEA model is that it allows each DMU to measure the efficiency using its favorable weights in order to calculate its maximum efficiency score [17]. With the flexibility of weights selection, a DMU would assign a larger multiplier (weight) to the dimension where it has a better performance. That is, a smaller input or a bigger output, compared with other DMUs, would be assigned a larger multiplier. In contrast, a bigger input or a smaller output would be assigned a smaller multiplier.

Denote () as the th input dimension. In the optimal solution of model (2) we can find that if performs better in input dimension , would then obtain a greater ratio of its weighted value to ’s total weighted input; that is, . Since the constraint is set in model (2), the ratio can be reduced to (equals to) .

However, sometimes model (2) encounters the existence of alternative weights, especially in the case of the extreme efficient units [2]. For example, if has the most excellent performance both in dimension and in dimension , either or would satisfy the constraints in model (2), without affecting ’s relative efficiency. To solve this problem, a two-stage process is developed as follows.

* Stage 1*. Solve model (2) to calculate efficiency score of .

* Stage 2*. Solve the following model to get the maximal value of :
where . The product of and could be interpreted as an index reflecting the utilization efficiency of input by .

The developments above can lead to the following definition to evaluate ’s performance in input dimension .

*Definition 1 (input dim-efficiency score, by Approach 1, input-oriented, under CRS assumption). *One defines the* input dim-efficiency score* of in dimension as the product of its maximal weighted value and its CCR efficiency score; that is, .

By the following definition we can distinguish DMUs’ performance in input dimension into two separate classes,* dim-efficient* and* dim-inefficient*.

*Definition 2 (input dim-efficiency, by Approach 1). *If , then is* dim-efficient* in input dimension ; if , then is* dim-inefficient* in input dimension .

Alternately, we turn to performance evaluation in output dimensions. Denote () as the th output dimension. Similar to input dimension, we can use a two-stage process to measure the efficiency on output . The first stage is the same, whereas the model in the second stage is different, which is now as follows, aiming to maximize the weighted value of to ’s weighted sum of outputs:
Denote and as the optimal solution of model (4); then ’s performance in output dimension can be assessed by the following definition.

*Definition 3 (output dim-efficiency score, by Approach 1, input-oriented, under CRS assumption). *One defines the* output dim-efficiency score* of in dimension as .

We can also distinguish between dim-efficient and dim-inefficient DMUs in output dimension by the following definition.

*Definition 4 (output dim-efficiency, by Approach 1). *If , then is dim-efficient in output dimension ; if , then is dim-inefficient in output dimension .

Now we have the following propositions.

Proposition 5. * and are unit-invariant; that is, they are independent of the units in which the inputs and outputs are measured.*

It holds because models in calculation process, (2), (3), and (4), are all units-invariant.

Proposition 6. * and are scale-invariant; that is, they are independent of the differences among DMUs’ production scales but depend only on input and output proportions, respectively.*

It holds since the evaluation result is based on ratio scores and not on the input/output numerical values.

Proposition 7. *Being CCR DEA efficient is a necessary condition for a DMU to be dim-efficient.*

*Proof. *Suppose is dim-efficient in input dimension ; then we have . Since , so , which indicates is CCR DEA efficient.

Similarly, suppose is dim-efficient in output dimension ; then we have . Since , so , which also indicates the CCR efficiency of .

A DMU’s being dim-efficient means that it has the most excellent performance in this dimension and should serve as a role model for dim-inefficient DMUs to imitate. On the contrary, a dim-efficient DMU fails to achieve a score of 1.0 because of the presence of another DMU that receives a higher ratio for the same set of weights.

Now we use a simple numerical example to illustrate the above definitions. Table 1 presents a data set for 6 DMUs with two inputs and a single output, along with their CCR efficiency scores from model (2) in the last row. In parenthesis are their dim-efficiency scores in the associated dimensions.