Abstract: Cost efficiency evaluates the ability of a decision making unit to produce the current outputs at minimal cost, given its input prices. In this study, we present an interpretation of the cost models and also we propose a new model to evaluate the cost efficiency measure of the decision making units when all input prices of each DMU are available. This model is more desirable than the traditional cost model in terms of complexity and the numbers of constraints and variables are reduced in the new proposed model. Moreover, the proposed model is compared with the standard DEA model to determine their relations. By this comparison an interesting relation between the cost efficiency and the technical efficiency is found.
INTRODUCTION
Performance evaluation is an important task for a Decision Making Unit (DMU) to find its weaknesses so that subsequent improvements can be made. Since, the pioneering work of Charnes et al. (1978), Data Envelopment Analysis (DEA) has demonstrated to be an effective technique for measuring the relative efficiency of a set of DMUs which utilize the same types of inputs to produce the same types of outputs. There are different types of efficiency. One of them is Cost Efficiency (CE). Cost efficiency evaluates the ability of a DMU to produce the current outputs at minimal cost, given its input prices.
The concept of CE can be traced back to Farrell (1957), who originated many of the ideas underlying efficiency assessment. The CE can be interpreted as a measure of the potential cost reduction achievable given the outputs produced and the current input prices at each DMU. Farrell s concept was further developed by Fare et al. (1985), who formulated a Linear Programming (LP) model for CE assessment. This LP model requires input and output quantity data as well as input prices at each DMU. Camanho and Dyson (2005) presented a weight-restricted DEA model for measuring the cost efficiency of DMUs and Jahanshahloo et al. (2008) simplified this version of the cost efficiency model.
In this study, we propose a model to evaluate the cost efficiency measure which is more desirable than the traditional cost model formulated by Fare et al. (1985) in terms of complexity and we compare the presented model with the CCR models (multiplier and envelopment forms).
DISCUSSION
Suppose that we have n DMUs with activity vectors (xtj, ytj ); j = 1, …, n, where xj and yj are nonnegative and nonzero column vectors in Um and Us, respectively. All DMUs, DMUj (j = 1,. . . , n), use the same number, m, of inputs (xij (i = 1,…, m)) to produce the same number, s, of outputs (yrj (r = 1, …, s)). Note that the input and output vectors of all DMUs are the same in type but different in quantity.
The CCR model, which was suggested by Charnes et al. (1978), is a fractional linear programming problem and can be solved by being transformed into an equivalent linear programming problem. This linear programming problem is the input-oriented multiplier CCR model and is as follows:
| (1) |
The dual of the above problem is the input-oriented envelopment CCR model and is as follows:
| (2) |
Farrell (1957) also proposed a measure of cost efficiency, which assumes that prices are fixed and known and maybe different among DMUs. Cost efficiency evaluates the ability of a decision making unit to produce the current outputs at minimal cost, given its input prices. In order to obtain a measure of cost efficiency for DMUs with multiple inputs and outputs, the minimum cost for the production of a DMU’s current outputs with existing input prices is obtained by solving the following linear problem, as first formulated by Fare et al. (1985):
| (3) |
In the above model, cio>0 is the price of input i of DMUo where o∈ {1, ..., n} is the index of the DMU under assessment. xi is a variable that, at the optimal solution, gives the amount of input i to be employed by DMUo in order to produce the current outputs at minimal cost, subject to the technological restrictions imposed by the existing productivity possibility set. Note that this model assumes that the input prices at each DMU are fixed and known, although they can differ between DMUs.
Cost efficiency is then obtained as the ratio of minimum cost with current prices (the optimal value of (3)) to the current cost at DMUo, as follows:
in which * indicates the optimality and it is clear that CEo≤1.
Because the costs of inputs are nonnegative for each DMU, there exists an optimal solution such as (λ*t, x*t) such that all constraints
i = 1, …, m, are binding on this solution, i.e.,
i = 1, …, m. This means that the cost efficiency of DMUo is obtained from the following model:
| (4) |
In modal (4) let Eq.
i = 1, ..., m, in the objective function of it, so modal (4) is equivalent to:
| (5) |
The dual of the above model is:
| (6) |
Now we compare Model (6) with the standard DEA model (Multiplier CCR model) to determine their relations.
Consider the multiplier CCR model (Model (1)). If in the multiplier CCR model we let
then this model is equivalent to Model (6). That is, in the multiplier CCR model, if variables vi are replaced with constants
i = 1, …, m, then the optimal value of the objective function is the cost efficiency of DMUo.
Here, we present another version of the cost model to evaluate the cost efficiency measure.
Consider Model (5). Let
Then, Model (5) is converted to the following model:
| (7) |
The optimal value of the above model is equal to the optimal value of the following model:
| (8) |
Note that at optimality of (8), constraint
is binding.
The above model is another version of the cost model and its optimal value is the cost efficiency of DMUo. In this model, there are n+1 variables and r+1 constraints (without non-negativity constraints), while the standard minimal cost model has n+m variables and m+r constraints (without non-negativity constraints); that is, the proposed model has m-1 less variables and m-1 less constraints than the standard cost model formulated by Fare et al. (1985). Therefore, the complexity of our proposed model to evaluate the cost efficiency measure is much less than the traditional version.
Also, it is clear that the feasible region of the input-oriented CCR envelopment form (Model (2)) is a subset of the feasible region of the proposed model. Therefore, the cost efficiency of a DMU is not greater than the technical efficiency of this DMU, i.e.,
With attention to Models (2) and (8), we observe that the cost efficiency of a DMU can be evaluated using the input-oriented envelopment CCR model such that the summation of the first m constraints
multiplied by cio for i = 1, …, m
is used instead of the first m constraints
i = 1, …, m.
Furthermore, the cost efficiency of a DMU with m inputs and s outputs, as defined at the beginning of this section, is evaluated by using input-oriented envelopment CCR model. Now we define n reduced DMUs Dj with activity vectors
where
and
j = 1, 2, …, n, are nonnegative real numbers and vectors in in U and Us , respectively.
The technical efficiency of DMU Do using Model (2) is the cost efficiency of DMUo, i.e..
CONCLUSIONS
There are different types of efficiency measures such as technical efficiency and cost efficiency. These measures are different in terms of efficiency score values, because each uses different information obtained from decision making units, e.g., technical efficiency only uses the quantities of inputs and outputs and cost efficiency uses the quantities of inputs, outputs and all input prices. Any efficiency measure that uses more information obtained from DMUs is more realistic than the other efficiency measures; but because not all DMUs in the real world have the same type of information, we cannot use all DEA models for certain types of information. For example, to obtain the cost efficiency measure of a DMU, we need not only all input and output quantities but also all input prices and if one of the input prices is not available, we cannot obtain the cost efficiency of DMUs, but we can obtain the technical efficiency of these DMUs. In this study, a new model has been suggested to evaluate the cost efficiency measures of DMUs. This model is more desirable than the minimal cost model to evaluate the cost efficiency measure formulated by Fare et al. (1985) in terms of complexity. Also the relations between the proposed model and the envelopment and multiplier CCR models have been discussed.