Multi-valued indicators in DEA in the presence of undesirable outputs: A goal-directed approach

2021-12-10TongYao

中国科学技术大学学报 2021年7期

Tong Yao

School of Management, University of Science and Technology of China, Hefei 230026, China

Abstract: The data envelopment analysis (DEA) is an important data-driven method for the performance evaluation and performance improvement of a set of peer decision making units (DMUs), involving multiple inputs and multiple outputs which are identified as performance indicators. However, some performance indicators, unlike conventional DEA models with one single value, may have more than one value because of different definitions or measurement standards referring to multi-valued indicators. In addition, the performance indicators reflect the current status of DMUs, which ignore the goals of decision-makers. We first propose two modified slacks-based DEA models to deal with multi-valued indicators and provide the Pareto-optimal solution in two common decision-making scenarios, namely the decentralized and centralized decision-making cases. Furthermore, we extend the models by incorporating with the goals of decision-makers to help the DMUs improve their performance and get close to the goals of decision-makers as much as possible. The slacks-based approaches and integration of goals enhance the discriminability of the models to DMUs and provide more practical improvement for some indicators. A case study of 22 cities in the Yangtze River delta region in China is used to illustrate the effectiveness and practicality of our proposed models.

Keywords: Data envelopment analysis, multi-valued indicators, goals, decentralized decision-making, centralized decision-making

1 Introduction

The data envelopment analysis (DEA), first proposed by Charnes et al[1], is a well-known non-parametric data-driven tool for building a composite index (e.g., performance, benchmarking) of a set of homogeneous DMUs consuming multiple inputs to produce multiple outputs[2，3]. As one of the most important evaluation tools, DEA has been developed rapidly in both theory and application over the past four decades[4，5]. People now pay more and more attention to the environment. The DEA is a widely used method on energy and environment, where it usually involves the undesirable outputs in the researches[6], such as air pollutants. The decision-makers usually prefer to a smaller amount of undesirable outputs and the ways to deal with the undesirable outputs are widely studied in existing literatures[7，8].

Furthermore, the decision-makers’ goals, which reflect their preferences for performance evaluation and improvement direction, also play a decisive role in the selection of performance indicators. The performance indicators are directly related with the results of performance evaluation in DEA methods. The multi-valued indicators under different standards refer to different decision-makers’ goals. In other words, selecting an appropriate value for a multi-valued indicator is inevitably influenced by decision-makers’ goals, which further affects the results of performance evaluation and improvements. Sales targets, target yield for corporations, and air pollutant concentration limits are common examples of decision-makers’ goals, seen as the expected level. Such goals significantly impact performance evaluation in real-world situations. However, traditional DEA methods focus on comparisons among peer DMUs to provide evaluation and benchmarks without considering the goals of decision-makers. To fill this gap, indirect and direct DEA based approaches taking into account the goals of decision-makers have been proposed[13-15]. The former approaches replace the decision-makers’ goals with other values such as utility. Lozano et al[16]propose a bargaining based DEA approach considering the utility instead of goals to improve the performance of inefficient DMUs, while the complicated calculation process limits its utilization. The latter approaches handle the decision-makers’ goals in a direct way. For example, Stewart[17]proposes a new DEA model which constructs new reference points with the goals of top managers as benchmarks. Azadi et al[18]apply a goal-directed benchmarking method for supplier selection. Ruiz and Sirvent[19]introduce a DEA method to generate strongly efficient targets which satisfy the requirement of minimum distance to the goals and to the current performance. Besides, the goals are often established to plan the improvement. However, there are some goals that cannot be achieved at current production situation, referring to overly high goals, and there are also some goals that are unambitious, which cannot effectively guide the improvement, referring to overly low goals[20]. The DEA targets provide the best practices[19]. Accordingly, we further incorporate the decision-makers’ goals into our approaches to handle the problem of multi-valued indicators.

The rest of the paper is organized as follows. In the next section, we provide preliminaries. Section 3 introduces our proposed models to deal with the problem of multi-valued indicators and the extended models considering the goals of decision-makers in decentralized and centralized decision-making cases. In Section 4, we apply our models to evaluate the environmental performance of the cities in the Yangtze River delta region in China. Finally, we conclude the paper and discuss further extensions.

2 Preliminaries

Table 1. Illustration of the notations.

(1)

We usethe following three vectorsx∈Rm,y∈Rsandb∈Rlto represent inputs, desirable outputs, and undesirable outputs. Constructed by the inputs and outputs of all DMUs, the production technologyTis defined as follows:

T={(x,y,b)|xcan produceyandb}

(2)

Denoted matricesX,YandBasX=[xij]=[x11,…,xmn]∈Rm×n,Y=[yrj]=[y11,…,ysn]∈Rs×nandB=[bfj]=[b11,…,xln]∈Rl×n. The technology set under variable returns to scale (VRS) is given as:T(x)={(y,b)|x≥λX,y≤λY,b≥λB,≥0,λTe=1}, where an vector representing intensity variable. The VRS assumption is a broader scenario in reality since the full proportionality assumption under CRS assumption is not often satisfied[21]. We deal with undesirable outputs following the strong disposability assumption[22]. The reason is that the implicit assumption of the weak disposability is that all DMUs use the same abatement factor, which is inconsistent with the practice of focusing emission reduction efforts on DMUs with less emission reduction costs[23]; some outputs are also inappropriate for weak disposability assumption like SO2emissions[24].

2.1 Modified slack-based model

According to the slacks-based measure (SBM), which is first proposed by Tone[25]and later extended to the situation with undesirable outputs. We use the following modified slacks-based model to measure the inefficiency of a specific DMUo.

(3.1)

(3.2)

(3.3)

(3.4)

(3)

2.2 Multi-valued measures selection based on DDF

maxβ

(4.1)

(4.2)

(4.3)

(4.4)

(4.5)

(4.6)

(4.7)

(4.8)

(4.9)

(4.10)

λj≥0,j∈J,i∈I,r∈R,f∈F

(4.11)

(4)

whereMis a large positive number. Denote theβ*as the optimal objective function value, whereβ*≥0. Constraint (4.5)ensuresβ*≤1. Model (4) ignores the slacks, which may not guarantee the Pareto optimal solution. Besides, it does not consider the inefficiency of inputs, thus over estimating the performance.

3 Methodology

In this section, we introduce two modified slacks-based models to handle the problems of multi-valued input/output indicators. Then we incorporate the decision-makers’ goals to extend the models. Our models consider decentralized decision-making cases and centralized decision-making cases.

3.1 Modified slacks-based models without goals in the presence of multi-valued indicators

3.1.1 Decentralized decision-making case without goals

(5.1)

(5.2)

(5.3)

(5.4)

(5.5)

(5.6)

(5.7)

(5.8)

(5.9)

(5.10)

(5.11)

(5.12)

r∈RM,f∈FM

(5.13)

(5.14)

(5)

(6)

Specifically, model (3) can be converted into model(7) to obtain the inefficiency score of DMUoby solving thekth(k=1,…,K)∈combination of multi-valued performance indicators.

(7)

ProofWe assume the optimal objective function value of model (5) is

Let the optimal solution of model (7) be thedth,d∈combination of the multi-valued indicators, that is}. The optimal solution of model (5) withf∈FMdreferring to the selected multi-valued indicators is also a feasible solution to model (7). We get thesince model (7) is a maximization problem.

Then we assume theIMc,RMc, andFMcrespectively indicate the set of selected multi-valued inputs, multi-valued desirable outputs, and multi-valued undesirable outputs in thecth,c∈combination.Assume the maximum optimal objective function value of model (7)}. The solution of model (7) incorporating withf∈FMcis also feasible solution to model (5). We getsince model (5) is maximization problem.

However, model (5) is a nonlinear programming problem. Model (5) can be converted into a linear programming model following the way of Cook et al[3]; The detailed transformation process is shown in Appendix A.

3.1.2 Centralized decision-making case without goals

In the centralized decision-making case,all DMUs are controlled by central decision-makers. The central decision-makers make decisions from an overall perspective rather than any individual DMU’s point of view. In other words, all DMUs are assessed by using one consistent standard on performance indicators, which is selected by the central decision-makers.

In the case of centralized decision-making, we want to achieve the overall maximum improvements with one consistent set of input/output indicators for all DMUs. The model (8) is introduced and is shown as

(8.1)

(8.2)

(8.3)

(8.4)

(8.5)

(8.6)

(8.7)

(8.8)

(8.9)

(8.10)

(8.11)

(8.12)

r∈RM,f∈FM

(8.13)

(8.14)

(8)

where

(9)

Note that each evaluated DMU selects its preferred input/output indicators in model (5), whereas model (8) uses the same performance indicators for all assessed DMUs. The different selected standards lead to different performance evaluation results for the DMUs. Here, we discuss the relationship between models (5) and (8).

The model (8) has an advantage in computation faced with multi-valued indicators, which selects suitable value for multi-valued indicators in one time calculation for all DMUs. Besides,model (8) can be similarly transformed into a linear programming model; Appendix B gives the details.

3.2 Modified slacks-based models with goals in the presence of multi-valued indicators

In real-world practice, the selection of inputs/outputs may depend on the decision-makers’ preferences[30]. Goals such as the five-year economic development plan in China, the expected sales in a company, and the concentration limit on PM 10 represent the preferences of decision-makers. Goals are often set in organizational planning, which should not be ignored in performance evaluation and improvement[7, 31,32]. As a result, the goals may affect the selection of the multi-valued indicators and thus influence the performance evaluation results. Therefore, the goals of decision-makers should be considered. However, the goals set by the decision-makers may be unachievable or unambitious in practice[19]. Besides, the established goals may not be on best practice frontier. The DEA method provides the best practice frontier, which can be seen as the benchmark for the inefficient DMUs[17]. Accordingly, the DEA method can be used to guide the target setting. Therefore, we extend the above proposed slacks-based models by incorporating decision-makers’ goals. To be specific, our proposed models consider the best practice benchmark and decision-makers’ goals at the same time, which aim to find targets on best practice frontier as close as possible to decision-makers’ goals for DMUs. In this section, we unfold from two cases, including decentralized decision-making and centralized decision-making, to illustrate the effect of decision-makers’ goals.

3.2.1 Decentralized decision-making case with goals

(10)

s.t.(5.1-5.9)

(11.1)

(11.2)

(11.3)

(11.4)

(11.5)

(11.6)

(11.7)

(11.8)

(11.9)

j∈J,i∈I,r∈R,f∈F,

(11.10)

(11)

(12)

In addition to improving the input/output, the results also guide the targets setting.

Note that model (11) is a nonlinear programming model. We transform model (11) into a linear programming model based on the way of Cook et al[3]; details are in Appendix C.

3.2.2 Centralized decision-making case with goals

(13)

It implies the gaps between the targets (projection points) and the goals for DMUt,t∈J.

s.t.(8.1-8.9)

(14.1)

(14.2)

(14.3)

(14.4)

(14.5)

(14.6)

(14.7)

(14.8)

(14.9)

(8.10-8.13)

j∈J,i∈I,r∈R,f∈F,

(14.10)

(14)

We use similar way to transform this nonlinear programming model to a linear programming model for the ease of calculation, giving details in Appendix D.

4 Application to cities in the Yangtze River delta (YRD) region

The Yangtze River delta (YRD) region, locating at the strategic hub of China’s “Belt and Road” plan, plays an important role in China’s economic development[33]. The strategic concept of “Yangtze River delta Integration” was first proposed in 1982 and becomes a national strategy. The cities in the YRD region carry out extensive cooperation and use the same rules and regulations for management in some field. Increasing attention is paid to energy conservation and environmental protection in China, and the cities in the YRD region are significant for China to convert to a green economy. Therefore, in this section, our proposed models are used on the environmental performance evaluation and improvement for the cities in the Yangtze River delta (YRD) region of China in 2017.

4.1 Dataset

Table 2. Input/output indicators

Table 3. Cities in Yangtze River delta.

Table 4. Descriptive statistics of the data.

The data is collected from the China Statistical Yearbook, Urban-level Statistical Yearbook, and Urban Environment Bulletin. Based on the YRD urban agglomeration development plan released in 2016, 26 cities are included in the YRD region. Because of data availability, Yancheng, Taizhou, Jiaxing, and Zhoushan are excluded. Table 3 presents the provinces/municipalities and their constituents. The data statistics description is reported in Table 4. There are gaps among the cities shown in the line “S.D.” of Table 4.

4.2 Results and analysis

On the one hand, under the decentralized decision-making case, the frequency of selected multi-valued indicators and the change of the selected combinations of multi-valued indicators, obtained from models (5) and (11), are respectively shown in Tables 5 and 6. We can identify the indicators that need improvements through the frequency of selection. The higher the frequency of being selected, the poorer performance of most cities on this indicator. For instance, the most frequently selected indicators is PM 10 (b1) (16 times) if considering the goals, which means most cities need pay more attention to the PM 10 (b1) in view of the current level and targets setting. On the other hand, in the centralized decision-making situation, all cities are evaluated with the same standards, i.e., they have the same selected combination of the multi-valued indicators, that is, {y2,b1} and {y2,b2} obtained from model (8) and model (14), respectively.

Table 5. Frequency of selected multi-valued indicators in decentralized case.

Table 6. Frequency of selected combinations of multi-valued indicators in decentralized case.

By solving models (5), (8), (11) and (14), the inefficiency scores of 22 cities are obtained and reported in Table 7. Considering the situation without the goals, we draw the following conclusions. Firstly, less than 60% of the cities are efficient, whose inefficiency score equal to zero. It means that some cities are still environmentally inefficient, which calls for more efforts and management strategies in environmental improvement in the YRD region. Secondly, some cities such as Changzhou, whose inefficiency score under the centralized decision-making case (0.2043) is lower than that under the decentralized decision-making case (0.2146). Besides, it is clear that the average environmental performance in the centralized decision-making case is lower than that in the decentralized decision-making case, as seen in the last row of Table 7; this result is consistent with Theorem 3.2, and implies that characteristics of some cities may be ignored in the centralized decision-making case. Thirdly, under the same decision-making case, there are inter-city gaps in the environmental performance in the YRD region, which is consistent with the initial judgment in our

Table 7.Inefficiencyscoresofthecities.

CitiesWithout goalsDecentralizedCentralizedWith goalsDecentralizedCentralizedNanjing0.00000.00000.42610.4288Zhenjiang0.00000.00000.63360.6336Yangzhou0.00000.00000.59750.6152Changzhou0.21460.20430.54390.5723Suzhou0.00000.00000.39700.4095Wuxi0.17160.14660.47580.4480Nantong0.10630.10630.51940.5194Hangzhou0.16810.16590.44200.4222Huzhou0.00000.00001.04501.0537Shaoxing0.19030.17750.57030.5764Ningbo0.14390.13880.45090.4842Jinhua0.00000.00000.88790.8879Taizhou0.00000.00000.74280.7651Shanghai0.00000.00000.45070.4509Hefei0.00000.00000.53880.5558Wuhu0.11330.10190.98090.9814Chuzhou0.14040.13971.76341.8052Maanshan0.21870.18071.72751.7468Tongling0.00000.00002.90843.0638Chizhou0.00000.00005.04545.0542Anqing0.00000.00001.94541.9770Xuancheng0.00000.00002.45712.4933Average0.06670.06191.16141.1793

descriptive analysis in Table 4. Take the decentralized decision-making situation as an example, where the largest inefficiency score is 0.2187 (Ma’anshan) and the lowest inefficiency score is 0 (such as Nanjing) in Table 7.

Considering the situation with goals, the environmental performance results obtained from models (11) and (14) are shown in columns 4-5 of Table 7. We conclude the following results to illustrate the effects of goals. Firstly, compared with the case without goals, the higher inefficiency scores mean worse environmental performance due to the additional constraints of the goals. However, using goals avoids the problem that the performance evaluation under the centralized decision-making case is always higher than that under the decentralized decision-making case without goals, e.g., Nanjing, Changzhou. Secondly,considering goals enhances the discriminability of the models on DMUs. For example, under the centralized decision-making situation, there are more obvious differences among cities, where the gap between the maximum and minimum inefficiency scores is 4.6447 and 0.2043 respectively, seen in Table 7. The environmental performance gaps among cities are better distinguished, seen in Figure 1. Cities with more advanced economies often have a better environmental performance, as exemplified by Nanjing, Suzhou, and Shanghai. Besides, some DMUs such as Hefei that are classified as efficient when there are no goals can be distinguished after considering the goals. Besides, Suzhou have the best environmental performance with the lowest inefficiency score, which may be attributed to impressive economic development, and its economic level ranks among the top in China. However, Chizhou, as a less developed city, has the worst environmental performance with the maximum inefficiency score, i.e., 5.0454 and 5.0542 respectively under decentralized and centralized decision-making cases. The developed cities often pay more attention to the environment, which accords with the reality of China.

Figure 1. Inefficiency scores for cities under centralized decision-making case.

We then discuss the differences across provinces according to the regional division in Table 3. First, under decentralized decision-making case, provincial capitals such as Nanjing and Hefei (excepting Hangzhou) are environmentally efficient. The reason for the low environmental performance of Hangzhou may be that more human activities increase pollution (undesirable outputs). In addition, the capital of provinces often has a better environmental performance. Hefei, the capital of Anhui, has abetter environmental performance compared with some cities in other provinces, such as inefficiency score 0.5558 versus

Table 8. Average inefficiency scores of 4 provinces/municipalities.

Yangzhou’s 0.6152, as seen in column 5 in Table 7. Second, unbalanced environmental performance exists in YRD region according the average inefficiency scores for different provinces, shown in Table 8. Taking the centralized decision-making case as an example, Shanghai has the best environmental performance, and Anhui has the worst environmental performance. The results are more direct, seen in Figure 1. The cities in the right part of the graph with large fluctuations belong to Anhui province. Compared with other provinces or municipalities in YRD region, the reason behind the situation may be less-advanced economy in Anhui, and the government pays more attention to economic development rather than the environment. This result is not find in the case without goals, which implies the important role of goals. Besides, there are gaps among internal cities in Anhui province. Therefore, Anhui province needs to improve its environmental performance and it is necessary to adopt environmental policies tailored to local conditions for different cities.

The environmental inefficiency scores indicate these cities have the potential to enhance the environmental performance by adjusting their input/output. We consider the situation with goals and the adjustments of input/output can be obtained from models (11) and model (14). Taking Ma’anshan as an example, the adjustments of input/output indicators are listed in Table 9. There are some differences in the selected indicators, which affects the degree of improvement of input/output indicators, such as the adjustments for PM 10 (b1) with 9.9 in decentralized decision-making case, and PM 2.5 (b2) with 0.62 in the centralized decision-making case. It means Ma’anshan needs to pay more attention to the governance of PM 10 (b1), which reflects the short board of Ma’anshan in environment. However, from the centralized decision-making case, the selected PM 2.5 (b2) is short board for most cities in YRD. In short, for the purpose of supporting decision-making, it is useful to comprehensively consider different decision scenarios and the effect of goals.

Table 9. Adjustments of input/output indicators for Ma’anshan.

[Note] “/” means unselected indicators.

Setting suitable goals is a common policy to guide the environmental governance. Take the cities Nanjing, Jinhua, Shanghai and Hefei as examples, which are identified as efficient DMUs under the case without goals and are respectively located in Jiangsu, Zhejiang, Shanghai, Anhui, corresponding to the regional division of Table 3. Unlike the situation without goals, considering goals distinguishes these DMUs, and it guides the targets setting for these cities, as shown in Table 10. For inputs and undesirable outputs, the less is the better. The negative adjustments show that the goals are unachievable based on current situation, and it is more reasonable to set a higher value of goals. The positive adjustments show that the lower value of goals are suggested. In light of Table 10, for example, we take the SO2emission (b4) as an example. The negative adjustments for Nanjing(-15.77) means that the goal of decision-makers on SO2emission(b4) is currently unreachable, and higher value of the goal is suggested. However, for Jinhua (2.23), the positive adjustment means that lower value of goal is suggested. The analysis results for desirable outputs are contrary since the more is the better. The negative adjustments mean the goals are too high to reach currently and lower value of goals are more appropriate; the positive adjustments imply that it suggests to set higher value of goals. For instance, negative adjustments for Nanjing, Jinhua, Hefei under the centralized decision-making case indicate that the goals on GVA(y2) need to be set lower value. However, the goals on GVA(y2) for Shanghai with a positive adjustment under the centralized decision-making case is overly low and higher value of goal is suggested.

Table 10. Adjustments of goals for four cities.

In conclusion, our approach evaluates the environmental performance and guides the improvement in the presence of multi-valued indicators, and further provide the insight in incorporating with decision-makers’ goals. Given the aforementioned analysis, the environmental performance of cities has room to improve in the YRD region; the cities need pay more attention to energy conservation and environmental protection policies, and prevent short-sighted goals which ignore the environmental protection. Besides, the decision-makers can keep the whole picture of the centralized and decentralized decision-making cases in mind to support decisions on the environmental protection. Furthermore, adjustments in the environmental policies should be in line with the situation in each city due to gaps among the YRD cities, and the cities need to learn from successful practice and strengthen inter-provincial cooperation to achieve an integrated development strategy according to the practice. Additionally, apart from the environmental protection regulations and laws, appropriate goals should also be correlated with environmental protection efforts.

[Note] “/” mean unselected indicator.

5 Conclusion

Unlike the traditional DEA methods considering single-valued performance indicators, in this work, we first propose two modified slacks-based models to select a suitable value for the multi-valued indicators. Then we further incorporate goals into the proposed models. Aiming for more practicality, our proposed models consider two cases of the decentralized and centralized decision-making. Using an empirical example of 22 cities in the Yangtze River delta region in China, we demonstrate the applicability and practicality of our models.

To sum up, our models make the following contributions to the literature on multi-valued indicators in DEA. First, the new models not only provide insight into performance measurement, but also guide adjustments on input/output indicators and targets setting. Secondly, our models with the goals demonstrate that the performance evaluation and improvements are more in line with the expectations of decision-makers. Third, comparing with the modified SBM model, our models are easier to deal with multi-valued indicators through one time calculation. Fourth, our models are practical considering the goals of decision-makers in both the decentralized and centralized decision-making cases, which provide multi-faceted support for decision-makers.

This study can be extended as follows. First, the goals in this work are virtual values calculated from the original inputs/outputs data set.The goals are usually set by the interaction among decision-makers, considering many factors such as polices, prior performance in practice and are usually established before the performance evaluation. Future research can use the actual existed goals in specific practices. Besides, our models guide the targets setting, and the projection points obtained from our models can be set as new goals for the next production period according to the practical need. Second, our models assume that all the DMUs and all inputs/outputs of each DMU have goals. In reality, some indicators have no explicit goals; for these indicators, our models can be modified by removing the corresponding constraints of these indicators on goals. Third, besides the environmental performance evaluation, the models can also be used in other fields based on the actual need.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (71601173, 71631006, 71991464, 71921001).

Conflictofinterest

The authors declare no conflict of interest.

Authorinformation

TongYaois currently a postgraduate student in the School of Management under the supervision of Prof. Yang Feng and Assoc. Prof. Ang Sheng at University of Science and Technology of China. Her research focuses on decision analysis.

AppendixA

We first propose the Proposition 1, which is useful for converting the model to a linear programming model. The Proposition 1 is as follows.

Proposition 1For each DMUj,j∈J, the following inequalities hold

(A.1)