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.
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 Section 2. 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).
To develop a SBM network DEA model for the efficiency measurement of the recycling production system, we suppose the following notation related to Figure 1. Denote , as the direct inputs , as the feedback inputs. , as the desired outputs, , as the undesired outputs to subsystem 1 of the -th DMU. The output are the final output of subsystem 1; while the undesired outputs will be fed into subsystem 2 as inputs, with other controllable inputs, i.e., , to produce undesirable outputs , and desirable outputs , of the j-th DMU, . The desirable outputs will be poured back to subsystem 1 and become part of inputs to the production process.
In this work, the intermediate variables , and the feedback variables are assumed to be all consumed. Taking into account the intermediate undesirable outputs, the DEA reference technology of the production system can be defined aswhere represents the intensity vector for weighting the inputs and outputs in the production system. Moreover, under the assumption that the undesirable outputs produced in the recycling system are freely disposable (7), the DEA reference technology of the recycling system can be defined aswhere 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.
Let , denote the slack variables associated with the inequalities in (1), and , denote the slack variables associated with the inequalities in (2), respectively. The process efficiencies, , and system efficiency, , of DMU 0 are defined in (4) and (5), respectively (20).where the weights and are prespecified and denote the related importance to the system with and . 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:where 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:where
For simplicity of the notations, it is assumed that the first input of subsystem 1, , and the first undesired output of subsystem 2, , are approximately known in this work. The fuzzy efficiency of a recycling production system with imprecise data and can be obtained by solving the following fuzzy linear program:
It is noted that the imprecise data and in (8) can be represented by fuzzy sets with corresponding membership functions ad , respectively.
Definition 1. A triangular fuzzy number can be represented by the triplet of real numbers with the corresponding membership function:Assumed that the fuzzy sets and are triangular fuzzy numbers. Letdenote the -cuts of the fuzzy numbers and , respectively, where is the support of the fuzzy set and is the support of the fuzzy set . The lower and upper bounds of and can be calculated asTo derive the fuzzy efficiency in (7), Zadeh’s extension principle is employed, which can be described aswhere and are the membership functions of and , respectively. According to (12), the upper bound of the -cut of can be calculated by solvingAlso, the lower bound of the -cut of can be obtained: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 in (11).To obtain the upper bound 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: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: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 data and of DMU j, , by triangular fuzzy numbers and , respectively. Step 2. Denote -cuts of fuzzy numbers and as , , respectively. Step 3. Calculate the lower/upper bounds of and as Step 4. Derive the fuzzy efficiency for DMU . Step 4.1. Obtain the lower bound, , of by solving the single-level optimization problem (15). Step 4.2. Obtain the upper bound, , of 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 the -cut of the fuzzy system efficiency is the weighted average of the lower/upper bounds of the -cuts of its fuzzy process efficiencies.
Proof. Let (,) be an optimal solution of (15). The lower bound of the fuzzy system efficiency, , and its process efficiencies, and , can be obtained asDefine the weights:The average of the weighted process efficiencies isThat is, the lower bound of the -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 -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 force , capital and total energy consumption are utilized as inputs to produce a desirable output. The GDP is considered as an appropriate measure of the desirable output and total wastes generated 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 in subsystem 1 are regarded as the input to be treated with the newly necessary resources , such as the investment in treatment of environmental wastes and pollution. is named the intermediate variable between the subsystems 1 and 2. Moreover, the intermediate variable can be converted into desired outputs and undesirable outputs in subsystem 2. The former, such as volume of recycled solid waste , volume of backfill , and energy recovery , will be fed back into subsystem 1 as a special type of inputs, and the latter, such as landfill and other and waste incinerated without energy recovery 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 , waste incinerated without energy recovery , and energy recovery , 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 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 being equal to 0.8.
From Table 2, we find that the lower/upper bounds of the -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.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by MOST 111-2221-E-415-004.