REVENUE CONGESTION: AN APPLICATION OF DATA ENVELOPMENT ANALYSIS

. Congestion is generally used in the economics and indicates a situation where a decrease (increase) in one or more inputs can increase (decrease) one or more outputs. In this paper, we introduce a new concept using data envelopment analysis, and call it revenue congestion. The new concept im- plies a situation where reduction in some inputs may result in an increase in revenue. This improvement in revenue is rather possible by a simultaneous increase and decrease in outputs due to a reduction in inputs. Then, we try to propose a method to distinguish the revenue congestion and identify its sources and amounts. To illustrate the use of the proposed method, an empirical application corresponding to 30 Iranian bank branches is provided. 16 branches evidence revenue congestion via the proposed approach. This identiﬁcation is very signiﬁcant because these branches can increase the revenue of their out- puts by eliminating the amounts of revenue congestion in each of their inputs. Moreover, it is found that an increase in all outputs is not always proﬁtable, but rather in some cases a decrease in some outputs and an increase in some other outputs can help the ﬁrms to make more proﬁts.


1.
Introduction. The concept of congestion which is a widely observed economic phenomenon, indicates a situation where inputs are overinvested. A typical example is the case where too many men in an underground coal mine may lead to congestion.
The topic of congestion was initially defined and developed by Färe and Svensson [10] in 1980. Afterwards, it was extended by Färe and Grosskopf [11] within the data envelopment analysis framework. They imposed the assumptions of weak and strong disposability on the production possibility set to identify the occurrence of congestion. However, their approach has some difficulties to treat congestion [8]. In addition, a slacks-based measure was proposed by Cooper et al. [6] that has some advantages compared to the previous method and could identify the congested inputs and the amount of congestion in each input. Moreover, Jahanshahloo and Khodabakhshi [14] introduced an input relaxation model for improving outputs, and accordingly, calculated the input congestion based on the proposed model.
In fact, the investigations into congestion within the DEA framework have received a great deal of consideration in the last few decades. Some of the other research discoveries in this area include Tone and Sahoo [26], wei and Yan [27], Sueyoshi and Sekitani [25], Noura et al. [21] and Zare-Haghighi et al. [28], just to name a few.
According to Cooper et al. [7], the usual understanding of congestion is that a decrease (increase) in one or more inputs results in an increase (decrease) in one or more outputs. From this viewpoint, the values and prices of outputs are not important. Nevertheless, consider a case in which reduction in some inputs may cause an increase in some outputs and a decrease in some other outputs, and the increased outputs are more valuable and have higher prices. In this case, reduction in some inputs may increase the total revenue of outputs. This problem which has not been considered in the DEA literature, persuade us to introduce a new concept to deal with such circumstances. The new concept which is called "revenue congestion", indicates a situation where reduction in some inputs can increase the total revenue of outputs, or conversely, where an increase in inputs can reduce the total revenue of outputs. This improvement in revenue is not necessarily acquired by increasing all outputs. Besides, it is achieved by a simultaneous increase and decrease in outputs due to a reduction in inputs. In this paper, we try to propose a method to deal with revenue congestion and identify its sources and amounts.
Note that traditional congestion occurs when outputs are reduced due to excessive amount of inputs. The new concept which we introduce implies a situation where overuse in inputs results in the reduction of the revenue. In fact, when a DMU is revenue congested, it can enhance its revenue by eliminating the excessive amount of inputs. That is why we have chosen the name of revenue congestion for the new concept.
we emphasize that in this discussion the prices of outputs are assumed to be known. If they were unknown, the common set of weight DEA models could be employed to estimate a set of fair and impartial prices for the outputs. As you know, when the output prices are known, it is possible to estimate the revenue efficiency of the firms. In this paper, another concept of production (i.e., congestion) in the presence of output prices is studied.
The rest of this paper is arranged as follows: In section 2, we review the twomodel approach of Cooper et al. [4] for measuring congestion. The topics of revenue efficiency and common set of weights with the corresponding DEA models are also presented in this section. Section 3 focuses on the new concept of revenue congestion and provides a plain definition to clarify the matter. Furthermore, a method is proposed in order to deal with this type of congestion. In section 4, the results of the mentioned models are presented and interpreted, regarding an empirical application corresponding to 30 Iranian bank branches. Summary and conclusions of the study are provided in section 5.

2.1.
Cooper's method. Here, we review the Cooper's approach for evaluating congestion. This approach was first presented by Cooper et al. [6] and was then examined on real data and refined by Brocket et al. [4]. This approach is a slacksbased method, and progresses in two stages. The first stage is directed to find a projection point for each DMU by means of the output-oriented version of BCC model. The output-oriented version has received more attention in DEA-based congestion measurement, because the input-oriented version may yield wrong results in this area [7]. In the second stage, the outputs are fixed to those of the projection point, and afterwards, the maximum amount of inputs that can be augmented to the projection's inputs are computed.
To provide the related DEA models, we assume that there are n observed firms and each firm is supposed to consume m inputs and produce s outputs. The jth firm is shown by the vector (x j , y j ), where x j = (x 1j , x 2j , ..., x mj ) ≥ 0, x j = 0, and y j = (y 1j , y 2j , ..., y sj ) ≥ 0, y j = 0. In Cooper's congestion approach [4], which was mentioned above, at first the following well known BCC model is solved: In this model, λ j (j = 1, ..., n) is the jth structural variable corresponding to the jth DMU, and the o index refers to the DMU under evaluation. Also, s − i and s + r are the slack variables for decrease and increase in the ith input and rth output, respectively. Besides, ε is a Non-Archimedean element and applies only in theory to avoid rewriting the constraints of the model. In fact, the two phase procedure is employed to manage this element.
Let an optimal solution of model (1) be (λ * , ϕ * , s − * , s + * ). Therefore, the DMU o is called strongly efficient if and only if ϕ * = 1, s − * = 0 and s + * = 0 hold for every optimal solution of the model. Otherwise, the DMU o can be projected on a strongly efficient point ( x o , y o ) via the following formulations: It is known that inefficiency is a necessary condition for the occurrence of congestion [7]. After identifying whether DMU o is inefficient, Cooper et al. [4] utilized the projection point on the right hand side of model (2) as follows: λ j = 1 , λ j ≥ 0 (j = 1, ..., n), This model calculates the maximum amount that can be added to the ith input of the projection point in order to remain in T N EW , where At last, Cooper's measure of the ith input congestion, which is denoted here by s c i , is defined as: 2.2. A common set of weights. As pointed out in the introduction, this paper assumes that the evaluated firms have known output prices. However, in actual cases, lack of information about technology often hampers the assessment of revenue maximization. To solve this problem, we employ the common set of weights (CSW) DEA models. These models evaluate all the DMUs at the same situation and provide a common set of fair and impartial weights for assessing all of them. The idea of finding a common set of weights was originally explored by Cook et al. [5] and Roll et al. [22]. Later, It was studied by many researchers such as Sinuany-Stern and Friedman [24], Jahanshahloo et al. [16], Liu and Peng [20], Amin and Toloo [1], and many other researchers.
In this paper, we use the Liu and Peng [20] method to obtain a set of common weights for the evaluated firms. The reasons is that their approach is very simple and easy to be understood and calculated.
The following multiple objective function provides a set of common weights for maximizing the efficiency of all DMUs: Where the vectors p = (p 1 , p 2 , ..., p s ) t and v = (v 1 , v 2 , ..., v m ) t are, respectively, the weights of outputs and inputs. Note that none of the DMUs' efficiency scores is allowed to exceed 1 [20], and hence, the above model can be written as: To achieve a linear problem, the ratio form of constraints can be rewritten in a linear form, and subsequently, the following transformation is applied: .., n). Consequently, the following linear problem [20] that provides a set of common weights for maximizing the efficiency of all DMUs is obtained: After solving the above model, the common weights of outputs is used as the output prices in the next section.
2.3. Revenue efficiency. Measurement of the revenue efficiency of the firms is one of the most important aspects in applied production analysis. This subject was first discussed by Farrell [13] and Debreu [9], and was then developed into implementable form by Färe et al. [12] using linear programming technologies. Since then, the measurement of revenue efficiency has been extended in many studies. See, e.g., Cooper et al. [6], Kuosmanen and Post [17,18], Jahanshahloo et al. [15], Lin [19], Aparicio et al. [2,3], Sahoo et al. [23] among others.
As mentioned before, in this paper we assume that the evaluated firms face known output prices and try to maximize their revenue. Suppose that p = (p 1 , p 2 , ..., p s ) ∈ R s >0 is the unit common price vector of outputs. The revenue efficiency of DMU o is achieved by solving the following LP problem:   λ j = 1 , λ j ≥ 0 (j = 1, ..., n), y r ≥ 0 (r = 1, ..., s). Let the optimal solution of model (7) be (λ * , y * ). Hence, the revenue efficiency of DMU o (RE o ) is computed by the following ratio: 3. Revenue congestion. In this section the new concept of "revenue congestion" is introduced and expanded in the DEA framework. Let p = (p 1 , p 2 , ..., p s ) be the predetermined vector of output prices. Multiplying each output by its respective price, the new output py is obtained for each DMU. In this case, revenue congestion occurs when reduction in one or more inputs can be associated with an increase in py (without worsening any other input), or inversely, when an increase in one or more inputs can be associated with a decrease in py (without improving any other input).
Note that the difference between traditional congestion and revenue congestion is that the revenue congestion indicates that reduction in some inputs may result in increasing some outputs and decreasing some other outputs, while the total revenue of outputs increases. Or similarly, an increases in some inputs may cause reducing some outputs and increasing some other outputs, while the total revenue of the outputs reduces. Nevertheless, in the traditional congestion reduction of inputs never cause reducing any output and also, increases in inputs never cause increasing any output. Now, To examine the status of revenue congestion for each DMU, consider model (8) as follows: Here, s + r1 and s − r2 are the slack variables for increase and decrease in the rth output, respectively. Simply, it can be verified that model (8) is always feasible and its optimal objective function value is bounded. In the following theorem, we prove that the optimal value of the objective function is not negative, either.
Proof. s + r1 = s − r2 = 0 (∀r), λ j = 0 (j = o), λ o = 1 is a feasible solution to model (8), and the value of the objective function for this feasible solution is zero. So, the optimal objective function value is always greater than or equal to zero.
It should be noticed that model (8) is solved for DMU o to see whether it could attain more revenue by means of increasing or decreasing some outputs. If this is not possible, or equally if z * = 0, then revenue congestion does not prevail at DMU o . Otherwise, if z * > 0, DMU o may show revenue congestion. Therefore, to capture the input revenue congestion and identify its amounts, model (9) This model computes the minimum amount that can be reduced from the ith input in order to acquire the optimal amounts of outputs of model (8). If r c i > 0, hence, r c i is the amount of revenue congestion in the ith input of DMU o . Otherwise, if r c i = 0, revenue congestion does not occur in the ith input of DMU o . Based on the above discussion, it can be established that a revenue efficient DMU never evidences revenue congestion. In the following theorem, we demonstrate this topic.
Theorem 3.2. Let DMU o be revenue efficient, thus revenue congestion is not present in its performance.
Proof. Assume that DMU o is revenue efficient and (λ * , y * ) is the optimal solution of model (7). Therefore, RE o = py o /py * = 1 and py o = py * .
Let ( λ, s + 1 , s − 2 ) be the optimal solution of model (8) for evaluating DMU o . We prove that the optimal value of the objective function of this model is zero. Suppose that z * = 0, then z * > 0. Simply, it can be verified that ( λ, y o + s + 1 − s − 2 ) is a feasible solution to model (7) and the objective function value of this feasible solution is while py * is the optimal objective function value of model (7) and this leads to a contradiction. Consequently, z * = 0, and DMU o does not show revenue congestion.
4. Numerical example. In this section, we apply the proposed method for assessing the revenue congestion on 30 Iranian bank branches. Table 1 exhibits the inputs and outputs which were considered for the evaluation. Moreover, the price of outputs computed by model (6) is presented in the third column of this Table. Here, we employ 3 inputs and 5 outputs which have been listed in Table 2.
The obtained results of the presented revenue congestion method are exhibited in table 3. The first column shows the optimal values of the objective function of model (8). It is necessary to mention that ten branches (12, 13, 16, 18, 19, 21, 22,    As it can be seen in table 3, 16 branches (1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 17, 23, 24, 26, 28 & 29) evidence revenue congestion. This identification is very significant because these branches can increase the revenue of their outputs by eliminating the amounts of revenue congestion in each of their inputs. z * is the amount of enhanced revenue that they could attain. Here, we report this status for two branches. For example, branch 5 can reduce its first input in amounts of r c 1 = 7.696. Therefore, its first and fourth outputs increase and its other outputs decrease, and as a result, its revenue enhances to z * = 27633120.115. Or, branch 24 can reduce its first and second inputs in amounts of r c 1 = 0.135 and r c 2 = 509585807.112. Accordingly, its third, fourth and fifth outputs increase and its other outputs decrease, and as a result, its revenue enhances to z * = 26996245.014.
From this Table, Cooper's approach recognizes 12 (3, 4, 5, 6, 8, 9, 10, 11, 15, 17, 26 & 28) congested branch. Although the two approaches do not produce the same results, however, 10 branches show both Cooper's congestion and revenue congestion. Only two branches (6 & 15) are congested in Cooper's method but not in the new proposed approach. 6 branches (1, 2, 7, 23, 24 & 29) evidence revenue congestion but not Cooper's congestion. It is because reduction in the inputs of these branches may result in increasing some of their outputs and decreasing some other outputs, however, in Cooper's congestion reduction of inputs never cause reduction of any output. Nevertheless, they display revenue congestion, since their total revenue increases.
From the above explanations, it can be concluded that an increase in all outputs is not always profitable, but rather in some cases a decrease in some outputs and an increase in some other outputs can help the firms to make more profits. This finding is an important feature of the new proposed method. 5. Summary and conclusion. This paper presented a new DEA approach that lets the DMUs improve their revenue not necessarily by increasing all outputs. This improvement in revenue is rather possible by a decision that results in simultaneous increase and decrease in outputs due to a reduction in inputs. This decision was described as "revenue congestion".
Congestion is generally used in the economics and indicates a situation where a decrease (increase) in one or more inputs can increase (decrease) one or more outputs. The difference between traditional congestion and revenue congestion is that the revenue congestion indicates that reduction in some inputs may result in increasing some outputs and decreasing some other outputs, while the total revenue of outputs increases. Or similarly, an increases in some inputs may cause reducing some outputs and increasing some other outputs, while the total revenue of the outputs reduces. Nevertheless, in the traditional congestion reduction of inputs never cause reducing any output and also, increases in inputs never cause increasing any output.
After reviewing Cooper's approach [4] for evaluating congestion and the concept of revenue efficiency, we tried to propose a method to distinguish the revenue congestion and identify its sources and amounts.
To illustrate the use of the proposed method, an actual example corresponding to 30 Iranian bank branches was provided. 16 branches evidenced revenue congestion via the proposed approach. Nevertheless, this identification is very significant because these branches can increase the revenue of their outputs by eliminating the amounts of revenue congestion in each of their inputs.
An important finding of the proposed approach is that an increase in all outputs is not always profitable, but rather in some cases a decrease in some outputs and an increase in some other outputs can help the firms to make more profits.