# Measurement of Congestion in the Simultaneous Presence of Desirable and Undesirable Outputs

Authors: H. Zare-Haghighi; M. Rostamy-Malkhalifeh; G. R. Jahanshahloo
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

The concept of congestion, which is mainly applied in
economics, refers to a situation where inputs are overinvested. Many
studies have focused on congestion measurement by means of data
envelopment analysis (DEA). However, most of the previous
investigations only considered the framework of desirable outputs.
In fact, firms in the real world unavoidably generate undesirable
outputs (such as pollutants or wastes) along with desirable outputs.
Therefore, a new scheme is required for measuring congestion in the
simultaneous presence of both desirable and undesirable outputs.
This paper develops a nonradial efficiency measure for including
undesirable outputs into the environmental performance. Based on the
proposed model, a new definition and a new approach are presented to
deal with congestion in the simultaneous presence of desirable and
undesirable outputs. Then, this paper uses the presented method to
study the pollutants (waste gas emission and waste discharge) of 31
administrative regions of China. The finding indicates that 7
industries pay attention to the reduction of their pollutants
accompanying improvement of their commercial targets. Consequently,
they do not show congestion in any input.


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## Body

## 1. Introduction

The concept of congestion, which is mainly applied in economics, refers to a situation where inputs are overinvested [1]. A typical example of congestion is the case where too many men in an underground coal mine may reduce the output of coal.The topic of congestion was initially defined and extended by the essay of Fare and Svensson [2] in 1980. Afterwards, it was examined by Fare and Grosskopf [3] within the data envelopment analysis framework. They imposed the assumptions of weak and strong disposability on the production possibility set to identify the evidence of congestion. Besides, their approach suffers from some weaknesses in treating congestion [4]. In addition, a slacks-based measure was presented by Cooper et al. [5] that has some strong points compared to the previous method and can identify the congested inputs and the amount of congestion in each input. Moreover, Jahanshahloo and Khodabakhshi [6] developed an input relaxation model for improving outputs and, accordingly, calculated the input congestion based on the proposed model. Wei and Yan [7] estimated congestion by the ratio of technical efficiency to pure technical efficiency. However, their approach shows only existence or nonexistence of congestion and cannot provide a value for measuring the amount of congestion in each input.Indeed, the investigations into congestion within the DEA framework have received considerable attention in the last few decades. Some of the other investigations in this field include Asgharian et al. [8], Flegg and Allen [9], Khodabakhshi [10], Sueyoshi and Sekitani [11], and Tone and Sahoo [12], to name a few.According to Cooper et al. [1], the common 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, all outputs are expected to rise. Outputs of this kind are called desirable outputs. However, an important issue is that, in the real world, undesirable outputs (such as pollutants or wastes) are unavoidably generated along with desirable outputs. Therefore, a new scheme is required for measuring congestion in the simultaneous presence of both desirable and undesirable outputs.In recent years, many researchers have been trying to model undesirable outputs within the DEA framework. A common treatment of undesirable outputs is to regard them as inputs and apply the traditional DEA models. Some of the works in this area include Dyson et al. [13], Hailu and Veeman [14], Dyckhoff and Allen [15], and Sueyoshi and Goto [16]. Nevertheless, this routine causes two problems. First, the free disposability principle between inputs and bad outputs implies that a finite amount of inputs can produce an infinite amount of undesirable outputs, while this is physically impossible [17]. Second, the free disposability principle does not recognize the relation between desirable and undesirable outputs [18].Seiford and Zhu [19] employed an alternative approach that first multiplies each undesirable output by −1 and then adds a big enough positive scalar to them in order to let all negative undesirable outputs be positive. Nevertheless, this approach is only valid under the variable returns to scale condition, and, furthermore, it does not reflect a rational production possibility set.Scheel [20] inverted the undesirable output values and treated them as desirable. This nonlinear transformation may change the efficiency frontiers, though, and hence result in erroneous efficiency scores.However, this subject was first methodically dealt with by Fare et al.’s publications [17, 21]. They thought of undesirable outputs as outputs and tried to incorporate them into production possibility set under a new axiom. Thus, they employed the weak disposability assumption, which had been introduced by Shephard [22], between good and bad outputs. Later, the study was extended in this direction by scholars such as Fare et al. [23], Zhou et al. [24], Kuosmanen [25], and Kousmanen and Podinovski [18].Although there are many papers in the DEA literature which discuss the theory and applications of congestion (see, e.g., [1, 4, 7–12, 26, 27] among others), studying this subject in the presence of undesirable outputs is very young. In existing paper only, Wu et al. [28] have studied congestion measurement considering both desirable and undesirable outputs. However, they chose the method of Seiford and Zhu [19] to address undesirable outputs, and, moreover, they applied Wei and Yan approach [7] and determined only existence or nonexistence of congestion.In this paper, we choose the Kuosmanen [25] technology to address the undesirable outputs. The reasons for choosing this technology have been clarified in Section 2. Employing the Kuosmanen technology, we present a nonradial efficiency measure that incorporates in both good and bad outputs. Then, following Cooper’s idea for measuring congestion, a new definition and a new approach are presented to deal with congestion in the simultaneous presence of desirable and undesirable outputs.The rest of this paper is arranged as follows. In Section2, we review the two-model approach of Cooper et al. [5] for congestion measurement and the Kuosmanen [25] technology for addressing undesirable outputs. Section 3 provides an efficiency measure that incorporates in both good and bad outputs. Thereafter, we focus on congestion in the simultaneous presence of both desirable and undesirable outputs and offer a plain definition. Furthermore, an approach is proposed in order to deal with this type of congestion. In Section 4, the results of the presented models are supplied and interpreted, regarding an empirical application corresponding to 31 administrative regions of China. The summary of the study and directions for future researches are provided in Section 5.


## 2. Preliminaries

The assumption in this paper is that there aren observed and comparable Decision Making Units (DMUs). The jth DMU, j∈{1,…,n}, is determined by the vector (xj,gj,bj), where xj=(x1j,x2j,…,xmj)∈Rm, xj≥0,xj≠0, is the vector of inputs, gj=(g1j,g2j,…,gsj)∈Rs, gj≥0,gj≠0, is the vector of desirable (good) outputs, and bj=(b1j,b2j,…,bhj)∈Rh, bj≥0, bj≠0, is the vector of undesirable (bad) outputs. Next, Cooper’s method for congestion evaluation [5] and the Kuosmanen technology [25] for addressing undesirable outputs are delineated.


### 2.1. Cooper’s Method

Here, we review Cooper’s approach for congestion evaluation, which considers only desirable outputs. This technique was first published by Cooper et al. [5] and was soon thereafter examined on real data and refined by Brockett et al. [26]. This approach, which is a slacks-based method, progresses in two stages. The first stage is directed to find a target point for each DMU via the output-oriented version of the BCC model. The output-oriented version has received more attention in congestion assessment because the input-oriented version may lead to wrong results in this area [1]. In the second stage, the outputs are fixed to those of the target point, and, afterwards, the maximum amount that can be added to the target’s inputs is computed.In Cooper’s method, at first, the following well-known BCC model is solved:(1)max⁡φ+ε(∑i=1m‍si-+∑r=1s‍sr+)s.t.∑j=1n‍λjxij+si-=xik(i=1,…,m),∑j=1n‍λjgrj-sr+=φgrk(r=1,…,s),∑j=1n‍λj=1,λj≥0(j=1,…,n),hhhhhhhhhhhhhhhsi-≥0(i=1,…,m),hhhhhhhhhhhhhhhsr+≥0(r=1,…,s).In Model (1), λj(j=1,…,n) is the jth structural variable associated with the jth DMU, and the k index refers to the DMU under evaluation. Also, si- and sr+ are, respectively, the slack variables for the decrease in the ith input and increase in the rth output. In addition, ε is a non-Archimedean element and applies only in theory in order to avoid rewriting the constraints of the model. Actually, the two-phase procedure is utilized to handle this element.Let an optimal solution of Model (1) be (λ*,φ*,s-*,s+*). Thus, the DMUk 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 DMUk can be projected on a strongly efficient target point (x^k,g^k) by the following formulations:
(2)x^k=∑j=1n‍λj*xj=xk-s-*,g^k=∑j=1n‍λj*gj=φ*gk+s+*.It is widely known that inefficiency is a necessary condition for the existence of congestion [1]. After recognizing whether DMUk is inefficient, Brockett et al. [26] employed the target point (2) on the right-hand side of Model (3) as follows:
(3)max⁡∑i=1m‍δi+s.t.∑j=1n‍λjxij-δi+=x^ik(i=1,…,m),∑j=1n‍λjgrj=g^rk(r=1,…,s),∑j=1n‍λj=1,λj≥0(j=1,…,n),hhhhhhhhi0≤δi+≤si-*(i=1,…,m).This model calculates the maximum amount that can be added to theith input of the target point in order to remain in TNEW, which is obtained by assuming weak input disposability. In this way, the nonzero slack cannot be associated with any input. TNEW is shown as follows:
(4)TNEW={(x,g)∣x=∑j=1n‍λjxj,g≤∑j=1n‍λjgj,∑j=1n‍λj=1,λj≥0;j=1,…,n}.Eventually, Cooper’s measure of theith input congestion, which is denoted here by sic, is defined as
(5)sic=si-*-δi+*(i=1,…,m).


### 2.2. The Kuosmanen Technology

The production technology is characterized by the setT={(x,g,b)∣xcanproduce(g,b)}. Consider the following principles which have been introduced in the DEA literature [17, 25] for incorporating undesirable factors into production technology.(A1)
Strong (free) disposability of inputs and good outputs. If(x,g,b)∈T, 0≤g′≤g, and x′≥x, then (x′,g′,b)∈T.(A2)
Weak disposability of good and bad outputs. If(x,g,b)∈T, 0≤θ≤1, then (x,θg,θb)∈T.(A3)
T is convex.Axiom (A2) recognizes the relation between good and bad outputs because the pollutants and wastes can be reduced in proportion to the reduction of good outputs. The multiplierθ used in this axiom is pointed out as the abatement factor [25].Kousmanen [25] applied n different abatement factors to deal with axiom (A2). He argued that the correct minimum extrapolation technology necessitates n distinctive abatement factors. Therefore, he employed distinctive abatement factors θj corresponding to each observed firm j=1,…,n and developed the following technology (TK):
(6)TK={(x,g,b)∣∑j=1n‍λjxj≤x,∑j=1n‍θjλjgj≥g,∑j=1n‍θjλjbj=b,∑j=1n‍λj=1,λj≥0,(j=1,…,n),0≤θj≤1,(j=1,…,n)∑j=1n‍}.Subsequently, Kuosmanen and Podinovski [18] proved that TK is indeed the only technology that contains all the observed firms and satisfies the minimum extrapolation principle of DEA under the mentioned axioms of (A1), (A2), and (A3).In addition, it should be noted thatTK is nonlinear, since θj is multiplied by λj. Nevertheless, Kuosmanen [25] stated that TK can be linearized as follows:
(7)λj=θjλj︸ηj+(1-θj)λj︸μj(j=1,…,n),TK={(x,g,b)∣∑j=1n‍(ηj+μj)xj≤x,∑j=1n‍ηjgj≥g,∑j=1n‍ηjbj=b,∑j=1n‍(ηj+μj)=1,ηj≥0,μj≥0,(j=1,…,n)∑j=1n‍}.


## 2.1. Cooper’s Method

Here, we review Cooper’s approach for congestion evaluation, which considers only desirable outputs. This technique was first published by Cooper et al. [5] and was soon thereafter examined on real data and refined by Brockett et al. [26]. This approach, which is a slacks-based method, progresses in two stages. The first stage is directed to find a target point for each DMU via the output-oriented version of the BCC model. The output-oriented version has received more attention in congestion assessment because the input-oriented version may lead to wrong results in this area [1]. In the second stage, the outputs are fixed to those of the target point, and, afterwards, the maximum amount that can be added to the target’s inputs is computed.In Cooper’s method, at first, the following well-known BCC model is solved:(1)max⁡φ+ε(∑i=1m‍si-+∑r=1s‍sr+)s.t.∑j=1n‍λjxij+si-=xik(i=1,…,m),∑j=1n‍λjgrj-sr+=φgrk(r=1,…,s),∑j=1n‍λj=1,λj≥0(j=1,…,n),hhhhhhhhhhhhhhhsi-≥0(i=1,…,m),hhhhhhhhhhhhhhhsr+≥0(r=1,…,s).In Model (1), λj(j=1,…,n) is the jth structural variable associated with the jth DMU, and the k index refers to the DMU under evaluation. Also, si- and sr+ are, respectively, the slack variables for the decrease in the ith input and increase in the rth output. In addition, ε is a non-Archimedean element and applies only in theory in order to avoid rewriting the constraints of the model. Actually, the two-phase procedure is utilized to handle this element.Let an optimal solution of Model (1) be (λ*,φ*,s-*,s+*). Thus, the DMUk 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 DMUk can be projected on a strongly efficient target point (x^k,g^k) by the following formulations:
(2)x^k=∑j=1n‍λj*xj=xk-s-*,g^k=∑j=1n‍λj*gj=φ*gk+s+*.It is widely known that inefficiency is a necessary condition for the existence of congestion [1]. After recognizing whether DMUk is inefficient, Brockett et al. [26] employed the target point (2) on the right-hand side of Model (3) as follows:
(3)max⁡∑i=1m‍δi+s.t.∑j=1n‍λjxij-δi+=x^ik(i=1,…,m),∑j=1n‍λjgrj=g^rk(r=1,…,s),∑j=1n‍λj=1,λj≥0(j=1,…,n),hhhhhhhhi0≤δi+≤si-*(i=1,…,m).This model calculates the maximum amount that can be added to theith input of the target point in order to remain in TNEW, which is obtained by assuming weak input disposability. In this way, the nonzero slack cannot be associated with any input. TNEW is shown as follows:
(4)TNEW={(x,g)∣x=∑j=1n‍λjxj,g≤∑j=1n‍λjgj,∑j=1n‍λj=1,λj≥0;j=1,…,n}.Eventually, Cooper’s measure of theith input congestion, which is denoted here by sic, is defined as
(5)sic=si-*-δi+*(i=1,…,m).


## 2.2. The Kuosmanen Technology

The production technology is characterized by the setT={(x,g,b)∣xcanproduce(g,b)}. Consider the following principles which have been introduced in the DEA literature [17, 25] for incorporating undesirable factors into production technology.(A1)
Strong (free) disposability of inputs and good outputs. If(x,g,b)∈T, 0≤g′≤g, and x′≥x, then (x′,g′,b)∈T.(A2)
Weak disposability of good and bad outputs. If(x,g,b)∈T, 0≤θ≤1, then (x,θg,θb)∈T.(A3)
T is convex.Axiom (A2) recognizes the relation between good and bad outputs because the pollutants and wastes can be reduced in proportion to the reduction of good outputs. The multiplierθ used in this axiom is pointed out as the abatement factor [25].Kousmanen [25] applied n different abatement factors to deal with axiom (A2). He argued that the correct minimum extrapolation technology necessitates n distinctive abatement factors. Therefore, he employed distinctive abatement factors θj corresponding to each observed firm j=1,…,n and developed the following technology (TK):
(6)TK={(x,g,b)∣∑j=1n‍λjxj≤x,∑j=1n‍θjλjgj≥g,∑j=1n‍θjλjbj=b,∑j=1n‍λj=1,λj≥0,(j=1,…,n),0≤θj≤1,(j=1,…,n)∑j=1n‍}.Subsequently, Kuosmanen and Podinovski [18] proved that TK is indeed the only technology that contains all the observed firms and satisfies the minimum extrapolation principle of DEA under the mentioned axioms of (A1), (A2), and (A3).In addition, it should be noted thatTK is nonlinear, since θj is multiplied by λj. Nevertheless, Kuosmanen [25] stated that TK can be linearized as follows:
(7)λj=θjλj︸ηj+(1-θj)λj︸μj(j=1,…,n),TK={(x,g,b)∣∑j=1n‍(ηj+μj)xj≤x,∑j=1n‍ηjgj≥g,∑j=1n‍ηjbj=b,∑j=1n‍(ηj+μj)=1,ηj≥0,μj≥0,(j=1,…,n)∑j=1n‍}.


## 3. The Proposed Models

### 3.1. An Efficiency Measure

This section attempts to provide an efficiency measure that incorporates in both desirable and undesirable outputs. For this purpose, the Russell efficiency measure [29] of DEA is employed. This measure includes all the inefficiencies that the model can identify, as described by Pastor et al. [29]. Here, we have used the Kuosmanen technology to address the undesirable outputs. The proposed measure, which is denoted by RMU, is as follows:
(8)RMU=min⁡(1/(m+h))(∑i=1mθi+∑f=1hγf)(1/s)∑r=1sφrs.t.(θ1x1k,…,θmxmk,φ1g1k,…,φsgsk,hhhhhhhhγ1b1k,…,γhbhk)t∈TKhhhhhhh0≤θi≤1(i=1,…,m),hhhhhhhhhhφr≥1(r=1,…,s),hhhhhhi0≤γf≤1(f=1,…,h).In Model (8), the constraints θi≤1, φr≥1, and γf≤1 are included in the model in order to see whether a DMU can be found to dominate DMUk. If this is not possible, that is, RMU=1, DMUk is environmentally efficient; otherwise, it is inefficient. Here, if xik=0(bfk=0), then the term θi(γf) is deleted from the objective function. Moreover, if grk=0, then it is replaced with a very small positive number which serves as a penalty. Using TK, the outcome model is
(9)RMU=min⁡(1/(m+h))(∑i=1mθi+∑f=1hγf)(1/s)∑r=1sφrs.t.∑j=1n‍(ηj+μj)xij≤θixik(i=1,…,m),hhhhhh∑j=1n‍ηjgrj≥φrgrk(r=1,…,s),hhhhhh∑j=1n‍ηjbfj=γfbfk(f=1,…,h),hhhhhh∑j=1n‍(ηj+μj)=1,ηj≥0,hhhhhhhhhhμj≥0(j=1,…,n),hhhhhhh0≤θi≤1(i=1,…,m),hhhhhhhhhhφr≥1(r=1,…,s),hhhhhhh0≤γf≤1(f=1,…,h).TheRMU measure incorporates all of the inefficiencies that the model can identify. Simply, it can be verified that all of the constraints of the first, second, and third groups are binding on the optimality of Model (9). In the following theorem, we demonstrate that RMU lies between zero and unity.Theorem 1.
Consider the following:
0
<
R
M
U
≤
1.Proof.
Sinceθi=1(∀i), φr=1(∀r), γf=1(∀f)μj=0, (∀j), ηj=0, (∀j≠k), and ηk=1 is a feasible solution to Model (9) with objective function value 1, RMU≤1. Moreover, θi≥0(∀i), φr≥1(∀r), and γf≥0(∀f), and, therefore, RMU≥0. Now, we only need to prove that RMU≠0. Suppose that RMU=0. This implies that θi=0(∀i) and γf=0(∀f). Consequently, the undesirable constraints yield ηj=0(∀j), and hence the desirable constraints lead to φrgrk≤0(∀r). At last, it is inferred that grk=0(∀r), while this is a contradiction. Consequently, RMU≠0.Note that Model (9) is a fractional programming problem; however, it can be transformed into an equivalent linear programming problem by using the Charnes-Cooper transformation. Holding (1/s)∑r=1sφr=1/t and multiplying each constraint by t, we thereafter let tηj=ηj′, tμj=μj′, tθi=θi′, tφr=φr′, and tγf=γf′, and thus the following linear problem is achieved:
(10)RMU=min⁡1m+h(∑i=1m‍θi′+∑f=1h‍γf′)s.t.∑j=1n‍(ηj′+μj′)xij≤θi′xik(i=1,…,m),hhhhhhhi∑j=1n‍ηj′grj≥φr′grk(r=1,…,s),hhhhhhh∑j=1n‍ηj′bfj=γf′bfk(f=1,…,h),hhhhhhh∑j=1n‍(ηj′+μj′)=t,hhhhhhhhηj′≥0,μj′≥0(j=1,…,n),hhhhhhhhhhi0≤θi′≤t(i=1,…,m),hhhhhhhhhhhhhhφr′≥t(r=1,…,s),hhhhhhhhhhh0≤γf′≤t(f=1,…,h),hhhhhh∑r=1s‍φr′=s,t≥0.As a result, we can easily obtainRMU via a linear problem for assessing the environmental performance of DMUk. Note that since t>0, the transformation is reversible. Let (η′*,μ′*,θ′*,φ′*,γ′*,t*) be an optimal solution of Model (10); consequently, (η*=η′*/t*,μ*=μ′*/t*,θ*=θ′*/t*,φ*=φ′*/t*,γ*=γ′*/t*) is an optimal solution of Model (9).The output-oriented version of the proposed measure can be defined by ignoring the reduction of inputs, that is, by omittingθi, in the objective function of Model (9). Subsequently, the efficiency score RMU(O) can be achieved as follows:
(11)RMU(O)=min⁡(1/h)∑f=1hγf(1/s)∑r=1sφrs.t.∑j=1n‍(ηj+μj)xij≤xik(i=1,…,m),hhhhhhh∑j=1n‍ηjgrj≥φrgrk(r=1,…,s),hhhhhhh∑j=1n‍ηjbfj=γfbfk(f=1,…,h),hhhhhhh∑j=1n‍(ηj+μj)=1,hhhhhhhiηj≥0,μj≥0(j=1,…,n),hhhhhhhhhhhhhφr≥1(r=1,…,s),hhhhhhhhhh0≤γf≤1(f=1,…,h).In a similar way, as described above, the equivalent linear program of the output-oriented Model (11) can be obtained as follows:
(12)RMU(O)=min⁡1h∑f=1h‍γf′s.t.∑j=1n‍(ηj′+μj′)xij≤txik(i=1,…,m),hhhhhhhh∑j=1n‍ηj′grj≥φr′grk(r=1,…,s),hhhhhhhh∑j=1n‍ηj′bfj=γf′bfk(f=1,…,h),hhhhhhhh∑j=1n‍(ηj′+μj′)=t,hhhhhhhhηj′≥0,μj′≥0(j=1,…,n),hhhhhhhhhhhhhhφr′≥t(r=1,…,s),hhhhhhhhhiii0≤γf′≤t(f=1,…,h),hhhhhhhh∑r=1s‍φr′=s,t≥0.


### 3.2. Congestion in the Simultaneous Presence of Desirable and Undesirable Outputs

As noted earlier, in traditional congestion, only desirable outputs are considered. Nevertheless, in the real world, undesirable outputs (such as pollutants or wastes) are unavoidably generated along with desirable outputs. In this subsection, we attempt to develop a new scheme for measuring congestion in the simultaneous presence of both desirable and undesirable outputs. The definition we present is as follows.Definition 2.
Congestion occurs in the performance ofDMUk=(xk,gk,bk) whenever a reduction in one or more inputs can increase one or more desirable outputs and decrease one or more undesirable outputs without worsening any other input, desirable output, or undesirable output. In contrast, whenever an increase in one or more inputs can decrease one or more desirable outputs and increase one or more undesirable outputs without improving any other input, desirable output, or undesirable output.Now we have tried to identify this type of congestion and determine its sources and amounts. For this purpose, first, the output-oriented version of the proposed measure is employed. This model determines the maximally possible amount of desirable outputs and the minimally possible amount of undesirable outputs that can be acquired byDMUk.In fact, Model (11) recognizes whether DMUk could achieve more desirable outputs and fewer undesirable outputs. If this is not possible, that is, if RMU(O)=1, then congestion does not prevail at DMUk. Otherwise, congestion may be present in the performance of DMUk. Therefore, to capture the input congestion and identify its sources and amounts, Model (13) is employed as follows:
(13)min⁡∑i=1m‍sics.t.∑j=1n‍(ηj+μj)xij=xik-sic(i=1,…,m),∑j=1n‍ηjgrj≥φr*grk(r=1,…,s),∑j=1n‍ηjbfj=γf*bfk(f=1,…,h),∑j=1n‍(ηj+μj)=1,ηj≥0,μj≥0(j=1,…,n),hhhhhhhhhhsic≥0(i=1,…,m).Fixing the outputs on the obtained optimal amounts, Model (13) computes the minimum amount that can be reduced from the ith input of DMUk in order to acquire these optimal amounts of outputs and also reach TKNEW.T
K
NEW, which is gained by changing the input inequalities of TK to input equalities, is the NEW technology corresponding to Kuosmanen technology and is as follows:
(14)TKNEW={(x,g,b)∣∑j=1n‍(ηj+μj)xj=x,∑j=1n‍ηjgj≥g,∑j=1n‍ηjbj=b,∑j=1n‍(ηj+μj)=1,ηj≥0,μj≥0,(j=1,…,n)∑j=1n‍}.After solving Model (13), if sic>0, then sic is the amount of congestion in the ith input of DMUk. Otherwise, if sic=0, congestion does not occur in the ith input of DMUk.


## 3.1. An Efficiency Measure

This section attempts to provide an efficiency measure that incorporates in both desirable and undesirable outputs. For this purpose, the Russell efficiency measure [29] of DEA is employed. This measure includes all the inefficiencies that the model can identify, as described by Pastor et al. [29]. Here, we have used the Kuosmanen technology to address the undesirable outputs. The proposed measure, which is denoted by RMU, is as follows:
(8)RMU=min⁡(1/(m+h))(∑i=1mθi+∑f=1hγf)(1/s)∑r=1sφrs.t.(θ1x1k,…,θmxmk,φ1g1k,…,φsgsk,hhhhhhhhγ1b1k,…,γhbhk)t∈TKhhhhhhh0≤θi≤1(i=1,…,m),hhhhhhhhhhφr≥1(r=1,…,s),hhhhhhi0≤γf≤1(f=1,…,h).In Model (8), the constraints θi≤1, φr≥1, and γf≤1 are included in the model in order to see whether a DMU can be found to dominate DMUk. If this is not possible, that is, RMU=1, DMUk is environmentally efficient; otherwise, it is inefficient. Here, if xik=0(bfk=0), then the term θi(γf) is deleted from the objective function. Moreover, if grk=0, then it is replaced with a very small positive number which serves as a penalty. Using TK, the outcome model is
(9)RMU=min⁡(1/(m+h))(∑i=1mθi+∑f=1hγf)(1/s)∑r=1sφrs.t.∑j=1n‍(ηj+μj)xij≤θixik(i=1,…,m),hhhhhh∑j=1n‍ηjgrj≥φrgrk(r=1,…,s),hhhhhh∑j=1n‍ηjbfj=γfbfk(f=1,…,h),hhhhhh∑j=1n‍(ηj+μj)=1,ηj≥0,hhhhhhhhhhμj≥0(j=1,…,n),hhhhhhh0≤θi≤1(i=1,…,m),hhhhhhhhhhφr≥1(r=1,…,s),hhhhhhh0≤γf≤1(f=1,…,h).TheRMU measure incorporates all of the inefficiencies that the model can identify. Simply, it can be verified that all of the constraints of the first, second, and third groups are binding on the optimality of Model (9). In the following theorem, we demonstrate that RMU lies between zero and unity.Theorem 1.
Consider the following:
0
<
R
M
U
≤
1.Proof.
Sinceθi=1(∀i), φr=1(∀r), γf=1(∀f)μj=0, (∀j), ηj=0, (∀j≠k), and ηk=1 is a feasible solution to Model (9) with objective function value 1, RMU≤1. Moreover, θi≥0(∀i), φr≥1(∀r), and γf≥0(∀f), and, therefore, RMU≥0. Now, we only need to prove that RMU≠0. Suppose that RMU=0. This implies that θi=0(∀i) and γf=0(∀f). Consequently, the undesirable constraints yield ηj=0(∀j), and hence the desirable constraints lead to φrgrk≤0(∀r). At last, it is inferred that grk=0(∀r), while this is a contradiction. Consequently, RMU≠0.Note that Model (9) is a fractional programming problem; however, it can be transformed into an equivalent linear programming problem by using the Charnes-Cooper transformation. Holding (1/s)∑r=1sφr=1/t and multiplying each constraint by t, we thereafter let tηj=ηj′, tμj=μj′, tθi=θi′, tφr=φr′, and tγf=γf′, and thus the following linear problem is achieved:
(10)RMU=min⁡1m+h(∑i=1m‍θi′+∑f=1h‍γf′)s.t.∑j=1n‍(ηj′+μj′)xij≤θi′xik(i=1,…,m),hhhhhhhi∑j=1n‍ηj′grj≥φr′grk(r=1,…,s),hhhhhhh∑j=1n‍ηj′bfj=γf′bfk(f=1,…,h),hhhhhhh∑j=1n‍(ηj′+μj′)=t,hhhhhhhhηj′≥0,μj′≥0(j=1,…,n),hhhhhhhhhhi0≤θi′≤t(i=1,…,m),hhhhhhhhhhhhhhφr′≥t(r=1,…,s),hhhhhhhhhhh0≤γf′≤t(f=1,…,h),hhhhhh∑r=1s‍φr′=s,t≥0.As a result, we can easily obtainRMU via a linear problem for assessing the environmental performance of DMUk. Note that since t>0, the transformation is reversible. Let (η′*,μ′*,θ′*,φ′*,γ′*,t*) be an optimal solution of Model (10); consequently, (η*=η′*/t*,μ*=μ′*/t*,θ*=θ′*/t*,φ*=φ′*/t*,γ*=γ′*/t*) is an optimal solution of Model (9).The output-oriented version of the proposed measure can be defined by ignoring the reduction of inputs, that is, by omittingθi, in the objective function of Model (9). Subsequently, the efficiency score RMU(O) can be achieved as follows:
(11)RMU(O)=min⁡(1/h)∑f=1hγf(1/s)∑r=1sφrs.t.∑j=1n‍(ηj+μj)xij≤xik(i=1,…,m),hhhhhhh∑j=1n‍ηjgrj≥φrgrk(r=1,…,s),hhhhhhh∑j=1n‍ηjbfj=γfbfk(f=1,…,h),hhhhhhh∑j=1n‍(ηj+μj)=1,hhhhhhhiηj≥0,μj≥0(j=1,…,n),hhhhhhhhhhhhhφr≥1(r=1,…,s),hhhhhhhhhh0≤γf≤1(f=1,…,h).In a similar way, as described above, the equivalent linear program of the output-oriented Model (11) can be obtained as follows:
(12)RMU(O)=min⁡1h∑f=1h‍γf′s.t.∑j=1n‍(ηj′+μj′)xij≤txik(i=1,…,m),hhhhhhhh∑j=1n‍ηj′grj≥φr′grk(r=1,…,s),hhhhhhhh∑j=1n‍ηj′bfj=γf′bfk(f=1,…,h),hhhhhhhh∑j=1n‍(ηj′+μj′)=t,hhhhhhhhηj′≥0,μj′≥0(j=1,…,n),hhhhhhhhhhhhhhφr′≥t(r=1,…,s),hhhhhhhhhiii0≤γf′≤t(f=1,…,h),hhhhhhhh∑r=1s‍φr′=s,t≥0.


## 3.2. Congestion in the Simultaneous Presence of Desirable and Undesirable Outputs

As noted earlier, in traditional congestion, only desirable outputs are considered. Nevertheless, in the real world, undesirable outputs (such as pollutants or wastes) are unavoidably generated along with desirable outputs. In this subsection, we attempt to develop a new scheme for measuring congestion in the simultaneous presence of both desirable and undesirable outputs. The definition we present is as follows.Definition 2.
Congestion occurs in the performance ofDMUk=(xk,gk,bk) whenever a reduction in one or more inputs can increase one or more desirable outputs and decrease one or more undesirable outputs without worsening any other input, desirable output, or undesirable output. In contrast, whenever an increase in one or more inputs can decrease one or more desirable outputs and increase one or more undesirable outputs without improving any other input, desirable output, or undesirable output.Now we have tried to identify this type of congestion and determine its sources and amounts. For this purpose, first, the output-oriented version of the proposed measure is employed. This model determines the maximally possible amount of desirable outputs and the minimally possible amount of undesirable outputs that can be acquired byDMUk.In fact, Model (11) recognizes whether DMUk could achieve more desirable outputs and fewer undesirable outputs. If this is not possible, that is, if RMU(O)=1, then congestion does not prevail at DMUk. Otherwise, congestion may be present in the performance of DMUk. Therefore, to capture the input congestion and identify its sources and amounts, Model (13) is employed as follows:
(13)min⁡∑i=1m‍sics.t.∑j=1n‍(ηj+μj)xij=xik-sic(i=1,…,m),∑j=1n‍ηjgrj≥φr*grk(r=1,…,s),∑j=1n‍ηjbfj=γf*bfk(f=1,…,h),∑j=1n‍(ηj+μj)=1,ηj≥0,μj≥0(j=1,…,n),hhhhhhhhhhsic≥0(i=1,…,m).Fixing the outputs on the obtained optimal amounts, Model (13) computes the minimum amount that can be reduced from the ith input of DMUk in order to acquire these optimal amounts of outputs and also reach TKNEW.T
K
NEW, which is gained by changing the input inequalities of TK to input equalities, is the NEW technology corresponding to Kuosmanen technology and is as follows:
(14)TKNEW={(x,g,b)∣∑j=1n‍(ηj+μj)xj=x,∑j=1n‍ηjgj≥g,∑j=1n‍ηjbj=b,∑j=1n‍(ηj+μj)=1,ηj≥0,μj≥0,(j=1,…,n)∑j=1n‍}.After solving Model (13), if sic>0, then sic is the amount of congestion in the ith input of DMUk. Otherwise, if sic=0, congestion does not occur in the ith input of DMUk.


## 4. Numerical Example

In this section, we apply the proposed method for assessing the congestion in 31 administrative regions of China. See the data set in Table1 which is adopted form Wu et al.’s paper [28]. These data have two inputs: the total investment in fixed assets of industry (TIFA) and the electricity consumption by industry (EC), one desirable output: the gross industrial output value (GIOV), and two undesirable outputs: the total volume of industrial waste gas emission (TWGE) and the total volume of waste water discharge (TWWD). For ease of comparison, we have named the industries D1 to D31, which are depicted in the second column of Table 1.Table 1
Data set of industry of China in 2010.
District
DMU
Input1
Input2
Desirable output
Undesirable out1
Undesirable out2
TIFA
EC
GIOV
TWGE
TWWD
Anhui
D1
9,121.829
1077.91
18,732
17,849
70,971
Beijing
D2
4,554.356
809.9
13,699.84
4,750
8,198
Chongqing
D3
5,049.258
626.44
9,143.55
10,943
45,180
Fujian
D4
6,534.803
1315.09
21,901.23
13,507
124,168
Gansu
D5
2,274.305
804.43
4,882.68
6,252
15,352
Guangdong
D6
11,903.36
4060.13
85,824.64
24,092
187,031
Guangxi
D7
5,166.135
993.24
9,644.13
14,520
165,211
Guizhou
D8
2,483.012
835.38
4,206.37
10,192
14,130
Hainan
D9
903.8264
159.02
1,381.25
1,360
5,782
Hebei
D10
11,737.07
2691.52
31,143.29
56,324
114,232
Heilongjiang
D11
5,019.085
747.84
9,535.15
10,111
38,921
Henan
D12
12,868.24
2353.96
34,995.53
22,709
150,406
Hubei
D13
7,276.638
1330.44
21,623.12
13,865
94,593
Hunan
D14
7,374.157
1171.91
19,008.83
14,673
95,605
Inner Mongolia
D15
6,831.416
1536.83
13,406.11
27,488
39,536
Jiangsu
D16
18,977.92
3864.37
92,056.48
31,213
263,760
Jiangxi
D17
6,696.149
700.51
13,883.06
9,812
72,526
Jilin
D18
6,313.748
576.98
13,098.35
8.240
38,656
Liaoning
D19
12,480.94
1715.26
36,219.42
26,955
71,521
Ningxia
D20
1,193.702
546.77
1,924.39
16,324
21,977
Qinghai
D21
789.5051
465.18
1,481.99
3,952
9,031
Shaanxi
D22
5,462.784
859.22
11,199.84
13,510
45,487
Shandong
D23
17,664.34
3298.46
83,851.4
43,837
208,257
Shanghai
D24
4,252.32
1295.87
30,114.41
12,969
36,696
Shanxi
D25
4,702.091
1460
12,471.33
35,190
49,881
Sichuan
D26
9,790.274
1549.03
23,147.38
20,107
93,444
Tianjin
D27
4,571.888
645.74
16,751.82
7,686
19,680
Tibet
D28
306.567
20.41
62.22
16
736
Xinjiang
D29
2,749.838
661.96
5,341.9
9,310
25,413
Yunnan
D30
4,024.972
1004.07
6,464.63
10,978
30,926
Zhejiang
D31
10,246.41
2820.93
51,394.2
20,434
217,426We utilized the general algebraic modeling system (GAMS) software for the calculations. Table2 displays the results of the presented models. The first column shows the results of the proposed measure which is computed by Model (10). D2, D6, D16, D23, D24, D27, and D28 are identified as the environmentally efficient industries and attain RMU=1. Thus, we can conclude that these 7 industries pay attention to the reduction of their pollutants accompanying improvement of their commercial targets.Table 2
Results of the proposed models.
District
DMU
RM
U
RM
U
(
O
)
s
1
c
s
2
c
Anhui
D1
0.455
0.317
4054.732
41.241
Beijing
D2
1
1
0
0
Chongqing
D3
0.380
0.205
1481.994
0
Fujian
D4
0.526
0.372
1144.784
135.603
Gansu
D5
0.331
0.231
0
418.298
Guangdong
D6
1
1
0
0
Guangxi
D7
0.262
0.133
611.779
183.340
Guizhou
D8
0.257
0.161
0
410.458
Hainan
D9
0.375
0.248
0
27.604
Hebei
D10
0.364
0.290
7343.447
1344.598
Heilongjiang
D11
0.386
0.237
798.638
0
Hainan
D12
0.495
0.433
6144.005
584.392
Hubei
D13
0.517
0.395
1914.956
163.486
Hunan
D14
0.467
0.322
2278.853
122.766
InnerMongolia
D15
0.319
0.186
2277.060
726.930
Jiangsu
D16
1
1
0
0
Jiangxi
D17
0.505
0.360
2412.607
0
Jilin
D18
0.604
0.516
2487.404
0
Liaoning
D19
0.652
0.635
7390.180
116.468
Ningxia
D20
0.150
0.047
0
361.478
Qinghai
D21
0.229
0.114
0
355.012
Shaanxi
D22
0.381
0.217
908.428
49.320
Shandong
D23
1
1
0
0
Shanghai
D24
1
1
0
0
Shanxi
D25
0.300
0.136
147.735
650.100
Sichuan
D26
0.444
0.350
4273.281
313.386
Tianjin
D27
1
1
0
0
Tibet
D28
1
1
0
0
Xinjiang
D29
0.287
0.162
0
187.446
Yunnan
D30
0.255
0.165
0
292.561
Zhejiang
D31
0.697
0.597
1851.267
312.372The second column depicts the results of the output-oriented measure which is computed by Model (12). It should be noticed that some industries (such as D7, D8, D15, D20, D21, D25, D29, and D30) obtain a very low RMU(O). As a result, these industries should be seriously concerned about their outputs. Especially, they should pay more attention to the reduction of their pollutants and wastes.The amount of congestion in each input, which is calculated by Model (13), is presented in the two last columns of Table 2. The 7 above-mentioned industries, which performed efficiently, do not show congestion in any input. The other industries evidence congestion in one or both of their inputs. For instance, congestion exists in the second input of D5. Therefore, D5 can reduce its electricity consumption in amounts of s2c=418.298 and, accordingly, its desirable output increases and its undesirable outputs decrease. Or D25 shows congestion in both of its inputs. Therefore, it can reduce its investment in amounts of s1c=147.735 and its electricity consumption in amounts of s2c=650.100 and, accordingly, its desirable output increases and its undesirable outputs decrease.


## 5. Summary and Conclusion

The concept of congestion, which is mainly applied in the economics, points out a situation where inputs are overinvested. Many researchers have studied this subject using data envelopment analysis (DEA). However, most of the previous investigations only considered the framework of desirable outputs (see, e.g., [6–8, 10–12]). In fact, firms in the real world unavoidably generate undesirable outputs (like pollutants or wastes) along with desirable outputs. Therefore, a new scheme is required for measuring congestion in the simultaneous presence of both desirable and undesirable outputs.In this paper, we briefly introduced the Kuosmanen [25] technology which is available in the DEA literature for modeling environmental performance under the weak disposability assumption of desirable and undesirable outputs.Then, we attempted to present a nonradial efficiency measure that incorporates in both desirable and undesirable outputs. Based on the proposed model, a new definition and a new approach are presented to deal with congestion in the simultaneous presence of desirable and undesirable outputs.Afterwards, the presented method is applied to study the pollutants (waste gas emission and waste discharge) of 31 administrative regions of China. The finding of this paper shows that 7 industries pay attention to the reduction of their pollutants accompanying improvement of their commercial targets. Consequently, they do not show congestion in any input. The other industries evidence congestion in one or both of their inputs. Reducing the amount of congestion in each input, these industries can enhance their desirable output and decrease their undesirable outputs.Possible future research is to study ranking of DMUs with undesirable outputs. Ranking efficient DMU is an important issue in DEA literature and many researchers have worked in this domain; see, for example, [30–38]. However, the research on this subject in the presence of undesirable outputs can be a future research agenda.



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*Source: 512157-2014-04-15.xml*