# A Slacks-Based Measure Approach for Efficiency Measurement of Recycling Production Systems with Imprecise Data

Authors: Huang-Chu Huang; Fung-Bao Liu; Cheng-Feng Hu
Journal: Journal of Mathematics
 (2022)

---

## Abstract

Resource scarcity and environmental degradation have made sustainable resource utilization and environmental protection worldwide. The development of the circular economy is considered an approach for more appropriate economic and environmental management. In this work, a slack-based measure approach was proposed for the assessment of recycling production systems with imprecise data. The two-level mathematical programming approach was employed for obtaining the lower and upper bounds of fuzzy efficiencies by transformed conventional single-level mathematical programs. The relationship between the fuzzy system efficiency and process efficiencies was explored. The proposed method was applied to assess the recycling production system of EU countries. Our results show that the performance of the production system in EU countries was superior to that of the recycling system, and the inefficiency of the whole system was attributed mainly to the recycling inefficiency.


---

## Body

## 1. Introduction

Increasingly tight resource constraints and severe environmental degradation have made the coordination between economic prosperity and environmental regulation a major issue of concern to sustain our living world. The rise of the circular economy (CE) is one of the most promising economic trends for sustaining the global ecological system and developing industrial innovation. To satisfy the sustainable economic growth mode, resources are used sustainably and cyclically for reducing pollution emissions and saving resources in a CE system (1). As a typical environment-economy linking system, it is important to improve the ecoefficiency of the CE system. Research approaches, such as life-cycle analysis (2), material flow analysis (3), and ecological footprint analysis (4), have been adopted for evaluating ecoefficiencies of enterprises and industries. A systematic literature review on the evaluation methods of CE performance assessment was presented recently (5). Most of these methods involve subjective factors while determining the weights of indicators. Data envelopment analysis (DEA) avoiding the subjectivity of weight determination and using programming solvers to obtain the efficiency of decision-making units (DMUs) has been an efficient tool for ecoefficiency assessment.DEA dealing with the performance measurement of production systems has made great progress in its theory and application research. In the case of real production, desirable outputs such as the real gross domestic product (GDP) are always accompanied by various undesirable outputs, e.g., industrial wastes. To deal with the undesirable outputs in efficiency research, conventional DEA models have been extended by directly treating the undesirable outputs as inputs (6) and introducing the directorial distance function (7) or data transformation function (8). The DEA models mentioned above are based on radial measures, which mainly concern the proportional reduction or enlargement of inputs/outputs, and neglect slacks in undesirable outputs. It may lead to an overestimation in the energy efficiency measurement (9). To address this limitation, Tone introduced the nonradial slacks-based measure (SBM), DEA, which is more discriminating than radial DEA models by permitting for nonproportional adjustment of undesirable outputs (10). From then on, the SBM DEA approach has been applied extensively for environment efficiency measurement (for example, see studies of (11–14)).In conventional DEA models, only the inputs consumed by and the outputs produced from the system are considered, and internal interactions are not taken into account in measuring efficiencies. When a system is composed of several components operating interdependently, it has been found that ignoring the operations of the components may produce efficiency measures that are misleading (15). The network DEA is extended from the conventional DEA, which considers the relation and dependencies between internal links so that the efficiencies are measured more appropriately. Researchers have utilized network DEA to evaluate the CE system with a closed-loop circular structure. Wu et al. (16) described a network DEA model with a feedback factor to assess China’s CE. Sun et al. (17) proposed a noncooperative game network DEA model to evaluate the CE system in China’s different provinces. Ding et al. (18) combined an extended Malmquist index model with cooperative game network DEA to identify the dominant factor of CE performance changes in the long term. Their pioneering work provides more rational and accurate DEA models for the performance assessment of CE. However, their models still have some drawbacks. For instance, Wu et al.’s model (16) is based on radial measures, which neglects slacks when dealing with undesirable outputs and may produce efficiency measures that are misleading. Sun et al.’s model (17) ignores recycled resources and is unable to present the closed-loop network structure of a CE system. Moreover, because many circular indicators are uncertain and are associated with data collection and judgment elicitation, certain fuzziness exists. Therefore, a data-driven decision support system for efficiency measurement must consider the fuzziness of the data involved in a CE system.In the present study, we took for the first time, the feedback factors as well as imprecise data into consideration for the efficiency measurement of a CE system under the framework of SBM network DEA. The two-level mathematical programming approach is employed for obtaining the lower and upper bounds of fuzzy efficiencies by transformed conventional one-level mathematical programs. The relationship between the fuzzy system efficiency and process efficiencies is explored. An empirical study on assessing the recycling production system of EU countries is included. The paper is organized as follows: a fuzzy SBM network DEA model for the efficiency measurement of a recycling production system with imprecise data is proposed in Section2. The two-level mathematical programming approach is employed for obtaining the fuzzy efficiencies. The study of assessing the recycling production system of EU countries is included in Section 3. Concluding remarks are provided in Section 4.


## 2. The SBM Network DEA for Efficiency Measurement of the Recycling Production System with Imprecise Data

A recycling production model, named the CE model featured that materials and energy needed by production can be extracted not only from the underground resources but also from the recycling and reusing of the emitted wastes, was proposed in early 1990 (19). In a simplified recycling production system, the overall system can be divided into two subsystems, namely, the production system and recycling system, as shown in Figure 1. The production system (subsystem 1) describes the main production process to produce desirable outputs and undesirable outputs with resource inputs. The recycling system (subsystem 2) illustrates the processes of dealing with the undesirable outputs from subsystem 1, especially the treatment and reusing of undesirable outputs. It should be noticed that the undesired outputs from subsystem 1 become one of the inputs to subsystem 2, namely, the intermediate variables, which will be converted into desired outputs and undesired outputs in subsystem 2. The undesired outputs in subsystem 2 will be emitted into the environment and the desired outputs in subsystem 2 will be fed back into subsystem 1 as a special type of input, which entirely shows a closed-loop feedback cycle for a recycling production system (16).Figure 1
The recycling production system.To develop a SBM network DEA model for the efficiency measurement of the recycling production system, we suppose the following notation related to Figure1. Denote Xmj1,m=1,2,…,M, as the direct inputs Ypj2g,p=1,2,…,P, as the feedback inputs. Yrj1g,r=1,2,…,R, as the desired outputs, Ydj1b,d=1,2,…,D, as the undesired outputs to subsystem 1 of the j-th DMU. The output Yrj1g are the final output of subsystem 1; while the undesired outputs Ydj1b will be fed into subsystem 2 as inputs, with other controllable inputs, i.e., Xlj2,l=1,2,…,L, to produce undesirable outputs Yhj2b,h=1,2,…,H, and desirable outputs Ypj2g,p=1,2,…,P, of the j-th DMU, j=1,2,…,J. The desirable outputs Ypj2g will be poured back to subsystem 1 and become part of inputs to the production process.In this work, the intermediate variablesYdj1b,d=1,2,…,D, and the feedback variables Ypj2g,p=1,2,…,P are assumed to be all consumed. Taking into account the intermediate undesirable outputs, the DEA reference technology of the production system can be defined as(1)Tp=x1,y2g,y1g,y1b:∑j=1nλj1Xmj1≤xm1,m=1,2,…,M,∑j=1nλj1Ypj2g=yp2g,p=1,2,…,P,∑j=1nλj1Yrj1g≥yr1g,r=1,2,…,R,∑j=1nλj1Ydj1b=yd1b,d=1,2,…,D,λj1≥0,j=1,2,…,n,where λ11,λ21,…,λn1 represents the intensity vector for weighting the inputs and outputs in the production system. Moreover, under the assumption that the undesirable outputs Yhj2b,h=1,2,…,H produced in the recycling system are freely disposable (7), the DEA reference technology of the recycling system can be defined as(2)Trec=y2,y2g,y1g,y2b:∑j=1nλj2Xlj2≤xl2,l=1,2,…,L,∑j=1nλj2Yhj2b≤yh2b,h=1,2,…,H,∑j=1nλj2Ydj1b=yd1b,d=1,2,…,D,∑j=1nλj2Ypj2g=yp2g,p=1,2,…,P,λj2≥0,j=1,2,…,n,where λ12,λ22,…,λn2 represents the intensity vector for weighting the inputs and outputs in the recycling system. According to (20), the linkage between the production system and recycling system is defined as follows to consider the continuity of activities in the two processes.(3)∑j=1nλj1Ydj1b=∑j=1nλj2Ydj1b=yd1b,d=1,2,…,D,∑j=1nλj1Ypj2g=∑j=1nλj2Ypj2g=yp2g,p=1,2,…,P.Letsm1−, sr1g+ denote the slack variables associated with the inequalities in (1), and sl2−, sh2b− denote the slack variables associated with the inequalities in (2), respectively. The process efficiencies, Ep0,Erec0, and system efficiency, Esys0, of DMU 0 are defined in (4) and (5), respectively (20).(4)Ep0=1−1/M∑m=1Msm1−/Xm011+1/R∑r=1Rsr1g+/Yr01g,Erec0=1−1/L∑l=1Lsl2−/Xl021+1/H∑h=1Hsh2b−/Yh02b,(5)Esys0=w11−1/M∑m=1Msm1−/Xm01+w21−1/L∑l=1Lsl2−/Xl02w11+1/R∑r=1Rsr1g+/Yr01g+w21+1/H∑h=1Hsh2b−/Yh02b,where the weights w1 and w2 are prespecified and denote the related importance to the system with w1,w2≥0 and w1+w2=1. According to (21), the weights in (5) can be derived from the observations afterwards and need not be specified beforehand. The SBM network DEA model for measuring the efficiency of a recycling production system with imprecise data can then be formulated as follows:(6)E˜0=min1−1/M∑m=1Msm1−/X˜m01+1−1/L∑l=1Lsl2−/X˜l021+1/R∑r=1Rsr1g+/Y˜r01g+1+1/H∑h=1Hsh2b−/Y˜h02b,s.t.∑j=1nλj1X˜mj1=X˜m01−sm1−,m=1,2,…,M,∑j=1nλj1Y˜rj1g=Y˜r01g+sr1g+,r=1,2,…,R,∑j=1nλj2X˜lj2=X˜l02−sl2−,l=1,2,…,L,∑j=1nλj2Y˜hj2b=Y˜h02b−sh2b−,h=1,2,…,H,∑j=1nλj1Y˜dj1b=∑j=1nλj2Y˜dj1b=Y˜d01b,d=1,2,…,D,∑j=1nλj1Y˜pj2g=∑j=1nλj2Y˜pj2g=Y˜p02g,p=1,2,…,P,λj1,λj2≥0,j=1,2,…,n,where E˜0 is the fuzzy efficiency of DMU 0. It is noted that the problem (6) is a fuzzy linear fractional program, which can be linearized to the following fuzzy linear program:(7)E˜0=mint−1M∑m=1Ms^m1−X˜m01+t−1L∑l=1Ls^l2−X˜l02s.t.t+1R∑r=1Rs^r1g+Y˜r01g+t+1H∑h=1Hs^h2b−Y˜h02b=1,∑j=1nλ^j1X˜mj1=tX˜m01−s^m1−,m=1,2,…,M,∑j=1nλ^j1Y˜rj1g=tY˜r01g+s^r1g+,r=1,2,…,R,∑j=1nλ^j2X˜lj2=tX˜l02−s^l2−,l=1,2,…,L,∑j=1nλ^j2Y˜hj2b=tY˜h02b−s^h2b−,h=1,2,…,H,∑j=1nλ^j1Y˜dj1b=∑j=1nλ^j2Y˜dj1b=tY˜d01b,d=1,2,…,D,∑j=1nλ^j1Y˜pj2g=∑j=1nλ^j2Y˜pj2g=tY˜p02g,p=1,2,…,P,t>0,s^m1−,s^l2−,s^r1g+,s^h2b−,λ^j1,λ^j2≥0,m=1,…,M,r=1,…,R,l=1,…,L,h=1,…,H,j=1,…,n,where s^m1−=tsm1−,s^l2−=tsl2−,s^r1g+=tsr1g+,s^h2b−=tsh2b−,λ^j1=tλj1,λ^j2=tλj2≥0,m=1,…,M,r=1,…,R,l=1,…,L,h=1,…,H,j=1,…,nFor simplicity of the notations, it is assumed that the first input of subsystem 1,X˜1j1, and the first undesired output of subsystem 2, Y˜1j2b, are approximately known in this work. The fuzzy efficiency of a recycling production system with imprecise data X˜1j1 and Y˜1j2b can be obtained by solving the following fuzzy linear program:(8)E˜0=mint−1Ms^11−X˜101+∑m=2Ms^m1−Xm01+t−1L∑l=1Ls^l2−Xl02,s.t.t+1R∑r=1Rs^r1g+Yr01g+t+1Hs^12b−Y˜102b+∑h=2Hs^h2b−Yh02b=1,∑j=1nλ^j1X˜1j1=tX˜101−s^11−,∑j=1nλ^j2Y˜1j2b=tY˜102b−s^12b−,∑j=1nλ^j1Xmj1=tXm01−s^m1−,m=2,3,…,M,∑j=1nλ^j1Yrj1g=tYr01g+s^r1g+,r=1,2,…,R,∑j=1nλ^j2Xlj2=tXl02−s^l2−,l=1,2,…,L,∑j=1nλ^j2Y˜hj2b=tY˜h02b−s^h2b−,h=2,…,H,∑j=1nλ^j1Ydj1b=∑j=1nλ^j2Ydj1b=tYd01b,d=1,2,…,D,∑j=1nλ^j1Ypj2g=∑j=1nλ^j2Ypj2g=tYp02g,p=1,2,…,P,t>0,s^m1−,s^l2−,s^r1g+,s^h2b−,λ^j1,λ^j2≥0,m=1,…,M,r=1,…,R,l=1,…,L,h=1,…,H,j=1,…,n.It is noted that the imprecise dataX˜1j1 and Y˜1j2b in (8) can be represented by fuzzy sets with corresponding membership functions μX˜1j1 ad μY˜1j2b, respectively.Definition 1.
A triangular fuzzy number can be represented by the tripletx˜=xl,xm,xu of real numbers with the corresponding membership function:(9)μx˜x=x−xlxm−xl,ifxl≤x≤xm,xu−xxu−xm,ifxm≤x≤xu,0,otherwise.
Assumed that the fuzzy setsX˜1j1=xjl,xjm,xju and Y˜1j2b=yjl,yjm,yju are triangular fuzzy numbers. Let(10)X1j1α=x1j1∈SX˜1j1|μX˜1j1x1j1≥α=X1j1αL,X1j1αU,Y1j2bα=y1j2b∈SY1j2b|μY˜1j2by1j2b≥α=Y1j2bαL,Y1j2bαU,denote the α-cuts of the fuzzy numbers X˜1j1 and Y˜1j2b, respectively, where SX˜1j1 is the support of the fuzzy set X˜1j1 and SY˜1j2b is the support of the fuzzy set Y˜1j2b. The lower and upper bounds of X1j1α and Y1j2bα can be calculated as(11)X1j1αL=xjl+αxjm−xjl,Y1j2bαL=yjl+αyjm−yjl,X1j1αU=xju−αxju−xjm,∀j=1,2,…,n,Y1j2bαU=yju−αyju−yjm,∀j=1,2,…,n.
To derive the fuzzy efficiencyE˜0 in (7), Zadeh’s extension principle is employed, which can be described as(12)μE˜0e=supx,yminμX˜1j1x1j1,μY˜1j2by1j2g,∀j=1,2,…,n|e=E˜0,where μX˜1j1 and μY˜1j2b are the membership functions of X˜1j1 and Y˜1j2b, respectively. According to (12), the upper bound of the α-cut of E˜0 can be calculated by solving(13)E0αU=maxx1j1αL≤xij1≤xij1αUy1j2bαL≤y1j2b≤y1j2bαU∀jmint−1Ms^11−x101+∑m=2Ms^m1−xmo1+t−1L∑l=1Ls^l2−xlo2s.t.t−1R∑r=1Rs^r1g+Yro1g+t+1Hs^12b−y102b+∑h=2Hs^l2−yho2b=1,∑j=1nλ^j1x1j1=tx101−s^11−,∑j=1nλ^j1Xmj1=tXm01−s^m1−,m=2,3,…,M,∑j=1nλ^j1Yrj1g=tYr01g+s^r1g+,r=1,2,…,R,∑j=1nλ^j2y1j2b=ty102b−s^12b−,∑j=1nλ^j2y1j2b=ty102b−s^12b−,∑j=1nλ^j2Yhj2b=tYh02b−s^h2b−,h=2,3,…,H,∑j=1nλ^j1Ydj1b=∑j=1nλ^j2Ydj1b=tYd01b,d=1,2,…,D,∑j=1nλ^j1Ypj2g=∑j=1nλ^j2Ypj2g=tYpo2g,p=1,2,…,P,t>0,s^m1−,s^l2−,s^r1g+,s^h2b−,λ^j1,λ^j2≥0,m=1,2,…,Mr=1,2,…,R,l=1,2,…,L,h=1,2,…,H,j=1,2,…n.
Also, the lower bound of theα-cut of E˜0 can be obtained:(14)E0αL=minxij1αL≤x1j1≤x1j1αUY1j2bαL≤y1j2b≤Y1j2bαU∀jmint=1Ms^11−x101+∑m=2Ms^m1−Xm01+t−1L∑l=1Ls^l2−Xlo2s.t.t+1R∑r=1Rs^r1g+Yr01g+t+1Hs^12b−y102b+∑h=2Hs^h2b−Yh02b=1,∑j=1nλ^j1x1j1=tx101−S^11−,∑j=1nλ^j1x1j1=tx101−s^11−,∑j=1nλ^j1Xmj1=tXm01−s^m1−,m=2,3,…,M,∑j=1nλ^j1Yrj1g=tYr01g+s^r1g+,r=1,2,…,R,∑j=1nλ^j2Xlj2=tX102−s^l2−,l=1,2,…,L,∑j=1nλ^j2y1j2b−ty102b−s^12b−,∑j=1nλ^j2y1j2b=tYh02b−s^h2b−,h=2,3,…,H,∑j=1nλ^j1Ydj1b=∑j=1nλ^j2Ydj1b=tYd01b,d=1,2,…,D,∑j=1nλ^j1Ypj2g=∑j=1nλ^j2Ypj2g=tYp02g,d=1,2,…,P,t>0,s^m1,s^l2−,s^r1g+,s^h2b−,λ^j1,λ^j2≥0,m=1,2,…,Mr=1,2,…,R,.=1,2,…,L,h=1,2,…,H,j=1,2,…,n.
Models (13) and (14) are two-level optimization programs. Since both inner and outer programs of (14) are minimization problems, the two-level optimization problem (14) can be reduced to the following single-level optimization program by replacing the lower and upper bounds of X˜1j1,Y˜1j2b in (11).(15)E0αL=mint−1Ms^11−x101+∑m=2Ms^m1−Xm01+t−1L∑l=1Ls^l2−Xl02s.t.t+1R∑r=1Rs^r1g+Yr01g+t+1Hs^12b−y102b+∑h=2Hs^h2b−Yh02b=1,∑j=1nλ^j1x1j1=tx101−s^11−,∑j=1nλ^j2y1j2b=ty102b−s^12b−,∑j=1nλ^j1Xmj1=tXm01−s^m1−,m=2,3,…,M,∑j=1nλ^j1Yrj1g=tYr01g+s^r1g+,r=1,2,…,R,∑j=1nλ^j2Xlj2=tXl02−s^l2−,l=1,2,…,L,∑j=1nλ^j2Yhj2b=tYh02b−s^h2b−,h=2,3,…,H,∑j=1nλ^j1Y˜dj1b=∑j=1nλ^j2Y˜dj1b=tY˜d01b,d=1,2,…,D,∑j=1nλ^j1Y˜pj2g=∑j=1nλ^j2Y˜pj2g=tY˜p02g,p=1,2,…,P,xjl+αxjm−xjl≤x1j1≤xju−αxju−xjm,∀j,yjl+αyjm−yjl≤y1j2b≤yju−αyju−yjm,∀j,t>0,s^m1−,s^l2−,s^r1g+,s^h2b−,λ^j1,λ^j2≥0,m=1,…,M,r=1,…,R,l=1,…,L,h=1,…,H,j=1,…,n.
To obtain the upper boundE0αU of the fuzzy system efficiency, the two-level optimization problem (13) is considered. It should be noted that the inner program of (13) has a dual, which can be formulated as follows:(16)E0=Maxθs.t.2θ−x101θ11∑m=2MXm01θm1−∑r=1RYr01gθr1g−∑l=1LXl02θl2−y102bθ12b,∑h=2HYh02bθh2b−∑d=1DYd01bθ¯d1b−∑p=1PYp02gθ¯p2g≤2,θm1≤−1MXm01,m=1,2,…,M,θl2≤−1LXl02,l=1,2,…,L,11+y102bθ+θ12b≤0,1RYr01gθ−θr1g≤0,r=1,2,…,R,11+Yh02bθ+θh2b≤0,h=2,3,…,H,xlj1θ11∑m=2MXmj1θm1+∑r=1RYrj1gθr1g+∑d=1DYdj1bθd1b+∑d=1DYdj1bθ¯d1b+∑p=1PYpj2gθp2g≤0,j=1,2,…,n,∑l=1LXlj2θl2+y1j2bθ12b+∑h=2HYhj2bθh2b−∑d=1DYdj1bθd1b−∑p=1PYpj2gθp2g+∑p=1PYpj2gθ¯p2g≤0,j=1,2,…,n.
When the objectives of inner and the outer optimization of (13) have the same direction of optimization, i.e., maximization, the two-level optimization problem (13) can be converted to the following one-level optimization problem:(17)E0αU=Maxθs.t. Constraintsin15xjl+αxjm−xjl≤x1j1≤xju−αxju−xjm,∀j,yjl+αyjm−yjl≤y1j2b≤yju−αyju−yjm,∀j.
Moreover, since model (16) is the dual of model (13), the associated variables in (13) can be calculated from the reduced cost and dual price of constraints in (17). The lower and upper bounds of the fuzzy system efficiency and process efficiencies can then be derived.
Based on the above discussion, a procedure for obtaining the fuzzy efficiencies of the recycling production system with imprecise data is described as follows.
The procedure of the proposed method is as follows:Step 1. Represent the imprecise dataX˜1j1 and Y˜1j2b of DMU j, j=1,2,…,J, by triangular fuzzy numbers xjl,xjm,xju and yjl,yjm,yju, respectively.Step 2. Denoteα-cuts of fuzzy numbers X˜1j1 and Y˜1j2b as X1j1α=X1j1αL,X1j1αU, Y1j2bα=Y1j2bαL,Y1j2bαU, respectively.Step 3. Calculate the lower/upper bounds ofX1j1α and Y1j2bα as X1j1αL=xjl+αxjm−xjl,∀j=1,2,…,n,Y1j2bαL=yjl+αyjm−yjl,∀j=1,2,…,n,X1j1αU=xju−αxju−xjm,∀j=1,2,…,n,Y1j2bαU=yju−αyju−yjm,∀j=1,2,…,n.Step 4. Derive the fuzzy efficiencyE˜j for DMU j=1,2,…,J.Step 4.1. Obtain the lower bound,EjαL, of Ejα by solving the single-level optimization problem (15).Step 4.2. Obtain the upper bound,EjαU, of Ejα by solving the one-level optimization problem (17).
To investigate the relationship between the resulting fuzzy system efficiency and process efficiencies, an important proposition is explored.Proposition 1.
The lower/upper bound of theα-cut of the fuzzy system efficiency is the weighted average of the lower/upper bounds of the α-cuts of its fuzzy process efficiencies.Proof.
Let (t∗,t∗,s^m∗1−,s^l∗2−,s^r∗1g+,s^h∗2b−,λ^j∗1,λ^j∗2,x1j∗1,y1j∗2b) be an optimal solution of (15). The lower bound of the fuzzy system efficiency, E0αL, and its process efficiencies, Ep0αL and Erec0αL, can be obtained as(18)E0αL=t∗−1/Ms^1∗1−/x10∗1+∑m=2Ms^m∗1−/Xm01+t∗−1/L∑l=1Ls^l∗2−/Xl02t∗+1/R∑r=1Rs^r∗1g+/Yr01g+t∗+1/Hs^1∗2b−/y10∗2b+∑h=2Hs^h∗2b−/Yh02b,Ep0αL=t∗−1/Ms^1∗1−/x101+∑m=2Ms^m∗1−/Xm01t∗+1/R∑r=1Rs^r∗1g+/Yr01g,Erec0αL=t∗−1/L∑l=1Ls^l∗2−/Xl02t∗+1/Hs^1∗2b−/y10∗2b+∑h=2Hs^h∗2b−/Yh02b.
Define the weights:(19)w1=t∗+1/R∑r=1Rs^r∗1g+/Yr01gt∗+1/R∑r=1Rs^r∗1g+/Yr01g+t∗+1/Hs^1∗2b−/y10∗2b+∑h=2Hs^h∗2b−/Yh02b,w2=t∗+1/Hs^1∗2b−/y10∗2b+∑h=2Hs^h∗2b−/Yh02bt∗+1/R∑r=1Rs^r∗1g+/Yr01g+t∗+1/Hs^1∗2b−/y10∗2b+∑h=2Hs^h∗2b−/Yh02b.
The average of the weighted process efficiencies is(20)w1Ep0αL+w2Erec0αL=w1t∗−1/Ms^1∗1−/x101+∑m=2Ms^m∗1−/Xm01t∗+1/R∑r=1Rs^r∗1g+/Yr01g+w2t∗−1/L∑l=1Ls^l∗2−/Xl02t∗+1/Hs^1∗2b−/y10∗2b+∑h=2Hs^h∗2b−/Yh02bt∗−1/Ms^1∗1−/x10∗1+∑m=2Ms^m∗1−/Xm01+t∗−1/L∑l=1Ls^l∗2−/Xl02t∗+1/R∑r=1Rs^r∗1g+/Yr01g+t∗+1/Hs^1∗2b−/y10∗2b+∑h=2Hs^h∗2b−/Yh02b=E0αL.
That is, the lower bound of theα-cut of the fuzzy system efficiency is the weighted average of the lower bounds of its fuzzy process efficiencies. A similar argument holds for the upper bound of the α-cut of fuzzy system efficiency.


## 3. An Application to Assessing the Recycling Production System of EU Countries

To assess the recycling production system of EU countries, the labour forceX1j1, capital X2j1 and total energy consumption X3j1 are utilized as inputs to produce a desirable output. The GDP is considered as an appropriate measure of the desirable output Yj1g and total wastes generated Yj1b as the undesirable output in the production system (subsystem 1). Since the recycling system (subsystem 2) reuses the used resources consumed in the production system, the undesirable outputs Yj1b in subsystem 1 are regarded as the input to be treated with the newly necessary resources Xj2, such as the investment in treatment of environmental wastes and pollution. Yj1b is named the intermediate variable between the subsystems 1 and 2. Moreover, the intermediate variable Yj1b can be converted into desired outputs Ypj2g and undesirable outputs Yhj2b in subsystem 2. The former, such as volume of recycled solid waste Y1j2g, volume of backfill Y2j2g, and energy recovery Y3j2g, will be fed back into subsystem 1 as a special type of inputs, and the latter, such as landfill and other Y1j2b and waste incinerated without energy recovery Y2j2b will become harmless discharge to the environment.The abovementioned variables are collected as the EU country data to measure the efficiency of recycling production system from waste statistics of EU countries and the world bank data in 2017 (22). Since some information, such as the investment in treatment of environmental wastes and pollution X102,X222,X252, waste incinerated without energy recovery Y2,202b, and energy recovery Y3,202g, are imprecise or estimated data, and the approach for obtaining the system and process efficiencies of the recycling production system with imprecise data developed in the previous section is employed. The data of the 28 EU countries are presented in Table 1, where the imprecise data are represented by fuzzy sets X˜102=8408,10927,11290,X˜222=402.9,425.5,478.6,X˜252=267.7,319.7,441.6,Y˜2,202b=0,110100,3734821,Y˜3,202g=0,3351400,334821. The membership functions of the imprecise data are constructed from other observations of the factors corresponding to the imprecise data by setting the smallest, median, and largest of the observations as the most pessimistic, most likely, and most optimistic values, respectively. The proposed method was coded using the Matlab software. Table 2 shows the fuzzy efficiencies of the system and processes for each EU country at α being equal to 0.8.Table 1
Data for assessing the recycling production system of EU countries.
DMUX11X21X31Y1gY1bX2Y12bY22bY12gY22gY32gBelgium5004109635363324600063150004224640182150526094442048552929160796941524Bulgaria40259889966220900120510053262113774023601228940624743431047630543Czechia53874875324880342002538032548942140292389698231256506220736788139114365359Denmark3024626421444949300209817017361103602545358910776857650408994682Germany43294696913216447498004000498395139725842555947407986017091404085106410855284539988852Estonia694521228183020024277641391571046418720752350174327132168661887076Ireland22489700311609691001548338711595176663397493116352150971209634873569252Greece49062263016702274006900931210456101405554225486722383403355929135014145383Spain2301624738582497372001289467825561691773096916301524782727890729903060462686172France303195398101471584320032346756710927 (p)89123199505299319911780549232433474607801751551681Croatia183310352663923800528055332526449278025248979558213309465091818Italy25584318657115930376001639973881032323307593254480688611294075758816749969663403006Cyprus61136101758361003384016371956477210353089279460834512836602Latvia100850193820262002531726865151445327299181524359287757017228886Lithuania1481809051083070066456891463762191112651152221754132754601238363239Luxembourg2811007940381054001012988413639468200944735202625224491356621366130Hungary468624195178652830015908526159544534135917379386134172858137346117665638Malta21226135834010019712535833840198729659376420291249134580Netherlands9050154915495175280014102799738276484656382123312495642938074501065450090Austria4535901872812749000612265831742809451363Y¯22b2264108063675844441Y¯32gPoland1839384961666522830018199064137250950061256389354983993072944036605469604251699Portugal520731296161142960014734417425.5 (e)5106975233546142640596720140172510178428819Romania8939431092228022900177557398508167028856661010134871452468177559190250668909Slovenia996785948753290055177874238252271428517433237663615032501526539590Slovakia276219021104183200010606352319.7 (p)5065888014865030424683214449836327674662989Finland2687514902524843400122869413751081636884753685989080842450557119580Sweden5245123749325905080014162219876010806010637228137771702475279694274083936645999United Kingdom33693437140133688438002772724741360110410809602735540574134572148352175093717948588194Available flags: p: provisional; e: estimated.Table 2
Theα-cuts of fuzzy efficiencies.
DMUSystem efficiencyProduction efficiencyRecycling efficiencyBelgium(0.8389, 0.8481)(0.9014, 0.9518)(0.7393, 0.7520)Bulgaria(0.5967, 0.6006)(0.9377, 0.9777)(0.2186, 0.2369)Czechia(0.6686, 0.7177)(0.9408, 0.9734)(0.3228, 0.4133)Denmark(0.8308, 0.8954)(0.9578, 0.9984)(0.7522, 0.7803)Germany(0.9882, 1.0000)(1.0000, 1.0000)(0.9434, 0.9920)Estonia(0.5248, 0.5850)(0.9546, 0.9897)(0.1908, 0.2013)Ireland(0.8997, 0.9133)(0.9872, 0.9994)(0.8076, 0.8169)Greece(0.6612, 0.6988)(0.9056, 0.9389)(0.4125, 0.4431)Spain(0.9159, 0.9352)(0.9485, 0.9833)(0.8715, 0.8823)France(0.9423, 0.9739)(0.9306, 0.9828)(0.9203, 0.9468)Croatia(0.9098, 0.9408)(0.9803, 1.0000)(0.8365, 0.8534)Italy(0.9306, 0.9698)(0.9379, 0.9885)(0.9341, 0.9466)Cyprus(0.4206, 0.4820)(0.9627, 0.9944)(0.1942, 0.2413)Latvia(0.5818, 0.6087)(0.9378, 0.9754)(0.2263, 0.2348)Lithuania(0.7970, 0.8223)(0.9572, 0.9973)(0.6243, 0.6420)Luxembourg(0.8163, 0.8382)(0.9207, 0.9712)(0.7021, 0.7164)Hungary(0.6631, 0.6929)(0.9467, 0.9927)(0.3578, 0.3739)Malta(0.7553, 0.7805)(0.9228, 0.9479)(0.5767, 0.5938)Netherlands(1.0000, 1.0000)(1.0000, 1.0000)(1.0000, 1.0000)Austria(0.6881, 0.7082)(0.9432, 0.9946)(0.4196, 0.4365)Poland(0.5881, 0.6177)(0.9558, 0.9963)(0.2413, 0.2520)Portugal(0.7116, 0.7539)(0.9338, 0.9747)(0.5247, 0.5913)Romania(0.5646, 0.5902)(0.9543, 0.9736)(0.2289, 0.2332)Slovenia(0.5630, 0.5877)(0.9634, 0.9835)(0.1908, 0.1968)Slovakia(0.6595, 0.6706)(0.9596, 0.9686)(0.3665, 0.3739)Finland(0.7943, 0.8139)(0.9905, 0.9989)(0.6003, 0.6143)Sweden(0.8367, 0.8688)(0.9727, 0.9836)(0.7265, 0.7392)United Kingdom(0.9591, 0.9998)(0.9868, 1.0000)(0.9503, 0.9821)From Table2, we find that the lower/upper bounds of the α-cuts in system efficiencies are less than one except for Germany and the Netherlands. It is also seen in the subsystem efficiencies of these two countries. Moreover, the lower and upper bounds of every fuzzy system efficiency are values between the lower and upper bounds of its process fuzzy efficiencies, respectively. The fuzzy efficiencies of the production and recycling systems are listed in the last two columns of Table 2. It shows that the fuzzy efficiencies of production system are higher than those of the recycling system, and there is a stronger relationship between the efficiency of recycling system and the whole system in most of the EU countries. For instance, the system efficiency score of Cyprus is a value in the interval (0.4206, 0.4820). This means that Cyprus is inefficient. Further investigation into the fuzzy efficiencies of the production system and recycling system of Cyprus shows that the production system performs better. The efficiency score of the production system is a value in the interval (0.9627, 0.9944). This means that the production system of Cyprus is almost efficient. However, the efficiency score of the recycling system of Cyprus is a value in the interval (0.1942, 0.2413). It can be concluded that the recycling system is less successful in converting the inputs to outputs and the source of inefficiencies of the whole system in Cyprus is mainly from its recycling process.To summarize the finding of this study, we have the following results:(a)
The overall system is efficient when both subsystems are efficient.(b)
The lower and upper bounds for every fuzzy system efficiency are values between the lower and upper bounds of its subsystem fuzzy efficiencies, respectively.(c)
The performance of the production system was superior to that of the recycling system.(d)
The inefficiency of the whole system was attributed mainly to the recycling inefficient.


## 4. Conclusions

A novel SBM network DEA model for measuring the efficiency of the recycling production system with imprecise data was proposed. The two-level mathematical programming approach was employed to obtain the lower and upper bounds of fuzzy efficiencies by transformed conventional one-level mathematical programs. The relationship between the fuzzy system efficiency and process efficiencies was explored. The proposed method was applied to assess the CE system of EU countries. Results show that the performance of the production system was superior to that of the recycling system, and the inefficiency of the whole system was attributed mainly to the inefficient recycling subsystem. To improve the performance of the recycling system, environmental protection projects and full use of environmental equipment should be encouraged and supported for raising the utilization efficiency of environmental treatment inputs. The proposed model can also be applied to assess the CE performance and promote the CE development in enterprises as well as industries.



---

*Source: 4909856-2022-07-13.xml*