Suitability and redundancy of non-homogeneous weight restrictions for measuring the relative efficiency in DEA
Introduction
In data envelopment analysis (DEA), weight restrictions of different types are often used to incorporate additional knowledge about a production process, understood in a broad sense, into available standard models [2]. In practice, identification of a suitable form of weight bounds, and the assessment of their numerical values, may be the main concern. It has been shown recently, however, that there is another major problem that must not be overlooked.
The fractional linear CCR model and its linear forms [6], [29] are used to measure the maximum relative efficiency of the assessed DMU jo. Their optimal solutions are regarded as the input and output weights that represent DMU jo in the best light in comparison with all the other DMUs in the observed group. This interpretation remains correct if additional homogeneous weight restrictions, including bounds on ratios of weights, are incorporated into the CCR model or its linear forms [8], [10], [34], [35].
It has recently been demonstrated that, in the presence of additional non-homogeneous weight restrictions, which includes absolute weight bounds, the CCR model and its linear forms may identify the maximum relative efficiency of DMU jo incorrectly [23], [25]. This happens because the CCR model and its linear forms do not actually maximise the relative efficiency of the assessed DMU. Thus, for example, the CCR model maximises the absolute efficiency of DMU jo. Without additional weight restrictions, this is, of course, equivalent to maximising the relative efficiency of DMU jo, but this is generally not so in the presence of additional weight bounds.
In this paper the investigation of the effects of non-homogeneous weight restrictions in DEA models is taken further. It is shown that certain types of non-homogeneous restrictions do not cause the observed errors. These types are different for the CCR model and its two linear forms. Such weight restrictions will be referred to as suitable for the maximisation of the relative efficiency.
An important implication of the obtained results concerns the use of absolute weight bounds. Below we show that the CCR model correctly assesses the relative efficiency of DMU jo if, simultaneously, no strictly positive lower bounds are imposed on any of the input weights and no upper bounds on the output weights. In the linear DEA model that maximises the total virtual output of the assessed DMU, no problem occurs if no upper bounds are imposed on any of the output weights. In the linear model that minimises the total virtual input of the assessed DMU, no problem occurs if no strictly positive lower bounds are imposed on any of the input weights.
Issues concerning redundancy of certain types of weight restrictions in DEA models are also considered. For example, it is shown that, if only lower bounds are imposed on the input weights in the CCR model, these have no effect on the maximum relative efficiency of the assessed DMU jo, and hence can be deleted. This result is correct, regardless of whether or what linear constraints are imposed on the output weights. Its validity also extends to the linear analogue of the CCR model that minimises the total virtual input of the assessed DMU.
This may be a surprising result, given the fact that incorporating very small lower bounds on all the input and output weights is a standard technique used to convert DEA models to linear programs and to separate the weights from zero values. The latter is often held to be a way of preventing dominated units scoring the maximum efficiency of 1 by assigning zero values to some of the inputs or outputs. The results proved in this paper and a simple example show that, if the desired effect is achieved and a dominated unit jo is no longer efficient, then this may be because the model fails to correctly identify the optimum weights, and not because the weights are separated from zero.
The structure of the paper is as follows. In Section 2, basic definitions, including the definitions of the absolute and relative efficiency, are formally introduced. In Section 3, examples of non-homogeneous weight restrictions and their relevance to DEA are discussed. An example is then considered in Section 4 that illustrates the problem occurring if weight restrictions of certain types are imposed. In Section 5, the types of suitable non-homogeneous weight restrictions in the CCR model are identified. In Section 6, a similar investigation concerns the use of weight restrictions in two linear analogues of the CCR model. In 7 Redundant weight restrictions in the CCR model, 8 Redundant weight restrictions in the linear models, certain types of weight restrictions are shown to be redundant in the CCR model and its linear analogues. Based on this, an example is considered in Section 9 that illustrates the redundancy of small lower bounds ε on the input weights in one of the linear models.
Section snippets
Definitions
Assume that the efficiency of DMU jo is to be assessed relative to a group of n DMUs J={1,2,…,n}, where jo∈J. Vector-columns Xj∈Rm and Yj∈Rs represent respectively the m inputs and s outputs of each DMU j∈J. The input and output vectors of DMU jo are denoted Xo and Yo, respectively.
In [6] it was assumed that all the inputs and outputs were strictly positive. This assumption was later relaxed in [9] to include zero inputs and outputs. Below we assume that all the inputs and outputs are
Non-homogeneous weight restrictions
In the CCR model (3) or its linear analogues , , additional linear weight restrictions may be imposed on output and input weights. These can be stated in the following matrix–vector form:In (7), A and C are respectively l1×s and l2×m matrices, b∈Rl1 and d∈Rl2, where l1 and l2 are non-negative integers. The case in which no constraints are imposed on output or input weights is included in (7). In this case, l1=0 or l2=0, respectively. Weight restrictions (7) are homogeneous if both
Side effects of weight restrictions
In the following example, weight bounds are incorporated into the CCR model (3) and its linear equivalent (5). Further analysis shows that the resulting models may not correctly identify the maximum relative efficiency of the assessed unit. Modifications of this example are also used to illustrate the theorems proved below. Example 1 Consider two DMUs, as defined by Table 1. In the basic model (3) or its linear analogue (5), both DMUs 1 and 2 are efficient. We now demonstrate the effect of weight
Suitable weight restrictions in the CCR model
Consider the CCR model (3) with additional weight restrictions (7):In model (12), notation is used to emphasise that the objective function is maximised subject to additional weight restrictions, or assurance regions (AR) [35].
Example 1 shows that model (12) may assess the maximum relative efficiency of DMU jo and the optimal weights incorrectly. There may be two reasons for this. First, the optimal weights (u∘,v∘
Suitable weight restrictions in linear DEA models
In this section we consider the use of weight restrictions (7) in model (5), which is one of the two linear analogues of the CCR model. Model (5) with additional weight restrictions (7) takes on the formIn model (16) and below, the superscript N in the notation is used to emphasise that the optimal value is sought in the model with an additional normalising equation vTXo=1.
Example 1 shows that model (16) may incorrectly
Redundant weight restrictions in the CCR model
In the previous section, we have identified sufficient conditions on weight restrictions (7) under which the CCR model correctly assesses the maximum relative efficiency of DMU jo. We prove below that, if all components of vector b are strictly positive, then the constraints Au⩽b are redundant, i.e. their presence in the model does not affect the maximum relative efficiency of DMU jo (see the formal definition of redundancy below). A similar statement is true for the case where all components
Redundant weight restrictions in the linear models
Linear model (5) can be viewed as the CCR model with the additional restrictions vTXo⩽1 and −vTX⩽−1 on the input weights that normalise the total virtual input of DMU jo by 1. Similarly, the second linear model (6) can be viewed as the CCR model with the additional restrictions on the output weights. Based on this observation, we can apply Theorem 3 to these two linear DEA models.
Let us refer to model (17) that explicitly maximises the relative efficiency of DMU jo on the feasible region of
Small lower bound ε
Consider incorporating the following inequalities into linear model (6):where ε is a small positive number, often taken equal to 0.001 or smaller. This is a widespread modelling and computational technique used to convert program (6) to a linear programming model and solve it in one step by preventing the weights from taking on zero values. See, for example, [3], [5], [15], [22], [32].
The use of the same small positive ε for the inputs and outputs measured on different
Conclusion
A simple example, considered in this paper, has illustrated a problem that may arise if non-homogeneous weight restrictions are incorporated into a DEA model. Specifically, in the CCR model and its linear analogues, the maximum relative efficiency of the assessed DMU jo may not be attained at a set of the input–output weights that maximise its absolute efficiency. In this case, the relative efficiency of DMU jo is underestimated and the interpretation of the optimal weights as the weights
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