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March  2022, 12(1): 79-91. doi: 10.3934/naco.2021052

## Measuring efficiency of a recycling production system with imprecise data

 1 Department of Applied Mathematics, National Chiayi University, Chiayi, Taiwan 2 Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu, Taiwan 3 Graduate Institute of Human Resource and Knowledge Management, National Kaohsiung Normal University, Kaohsiung, Taiwan

* Corresponding author: Cheng-Feng Hu

Received  March 2020 Revised  October 2021 Published  March 2022 Early access  November 2021

Fund Project: The first author is supported by MOST grant 109-2221-E-415-010

Resources scarcity and environmental degradation have made sustainable resource utilization and environmental protection worldwide. A circular economy system considers economic production activities as closed-loop feedback cycles in which resources are used sustainably and cyclically. Improving the eco-efficiency of the circular economy system has both theoretical value and practical meaning. In this work, the efficiency measurement model of the circular economy system with imprecise data based on network data envelopment analysis is proposed. The two-level mathematical programming approach is employed for measuring the system and process efficiencies. The lower and upper bounds of the efficiencies scores are calculated by transformed conventional one-level linear programs so that the existing solution methods can be applied. The proposed method is applied to assess the circular economy system of EU countries. Our results show that most countries have large difference among fuzzy efficiencies between the production efficiency and recycling efficiency stages, which reveals the source that causes the low efficiency of the circular economy system.

Citation: Cheng-Feng Hu, Hsiao-Fan Wang, Tingyang Liu. Measuring efficiency of a recycling production system with imprecise data. Numerical Algebra, Control and Optimization, 2022, 12 (1) : 79-91. doi: 10.3934/naco.2021052
##### References:
 [1] L. Castelli, R. Pesenti and W. Ukovich, DEA-like models for the efficiency evaluation of hierarchically structured units, Eur. J. Oper. Res., 154 (2004), 465-476.  doi: 10.1016/S0377-2217(03)00182-6. [2] A. Charnes and W. W. Cooper, Programming with linear fractional functionals, Nav. Res. Logist. Q., 9 (1962), 181-186.  doi: 10.1002/nav.3800090303. [3] A. Charnes, W. W. Cooper and E. Rhodes, Measuring the efficiency of decision making units, Eur. J. Oper. Res., 2 (1978), 429-444.  doi: 10.1016/0377-2217(78)90138-8. [4] Y. Chen, J. Du, S. H. David and J. Zhu, DEA model with shared resources and efficiency decomposition, Eur. J. Oper. Res., 207 (2010), 339-349.  doi: 10.1016/j.ejor.2010.03.031. [5] W. Chen, W. J. Liu, Y. Geng, S. Ohnishi, L. Sun, W. Y. Han, X. Tian and S. Z. Zhong, Life cycle based emergy analysis on China's cement production, J. Clean. Prod., 131 (2016), 1-8. [6] W. D. Cook and L. M. Seiford, Data envelopment analysis (DEA)-Thirty years on, Eur. J. Oper. Res., 192 (2009), 1-17.  doi: 10.1016/j.ejor.2008.01.032. [7] R. F$\ddot{a}$re and S. Grosskopf, Network DEA, Socio. Econ. Plann. Sci., 4 (2000), 35-49. [8] C. Kao and S. T. Liu, Fuzzy efficiency measures in data envelopment analysis, Fuzzy Sets Syst., 113 (2000), 427-437. [9] H. Mikulcic, H. Cabezas, M. Vujanovic and N. Duic, Environmental assessment of different cement manufacturing processes based on emergy and ecological footprint analysis, J. Clean. Prod., 130 (2016), 1-25. [10] D. Pearce and R. K. Turner, Economics of Natural Resources and the Environment, The Johns Hopkins University Press, Baltimore, 1998. [11] B. Simon, What are the most significant aspects of supporting the circular economy in the plastic industry?, Resources, Conserv. Recy., 141 (2019), 299-300. [12] L. Sun, H. Li, L. Dong, K. Fang, J.Z. Ren, Y. Geng, M. Fujii, W. Zhang, N. Zhang and Z. Liu, Eco-benefits assessment on urban industrial symbiosis based on material flows analysis and emergy evaluation approach: a case of Liuzhou city, China. Resources, Conserv. Recy., 119 (2017), 78-88. [13] H. Wu, Y. Liu, Q. Xia and W. Zhu, Measuring efficiency of recycling systems based on data envelopment analysis (DEA) network: A case from Chinese provincial circular economy, Environ. Eng. Manag. J., 13 (2014), 1089-1099. [14] L. A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets Syst., 1 (1978), 3-28.  doi: 10.1016/0165-0114(78)90029-5. [15] H. Z. Zimmermann, Fuzzy Set Theory and Its Applications, Kluwer-Nijhoff, Boston, 1996. doi: 10.1007/978-94-015-7153-1.

show all references

##### References:
 [1] L. Castelli, R. Pesenti and W. Ukovich, DEA-like models for the efficiency evaluation of hierarchically structured units, Eur. J. Oper. Res., 154 (2004), 465-476.  doi: 10.1016/S0377-2217(03)00182-6. [2] A. Charnes and W. W. Cooper, Programming with linear fractional functionals, Nav. Res. Logist. Q., 9 (1962), 181-186.  doi: 10.1002/nav.3800090303. [3] A. Charnes, W. W. Cooper and E. Rhodes, Measuring the efficiency of decision making units, Eur. J. Oper. Res., 2 (1978), 429-444.  doi: 10.1016/0377-2217(78)90138-8. [4] Y. Chen, J. Du, S. H. David and J. Zhu, DEA model with shared resources and efficiency decomposition, Eur. J. Oper. Res., 207 (2010), 339-349.  doi: 10.1016/j.ejor.2010.03.031. [5] W. Chen, W. J. Liu, Y. Geng, S. Ohnishi, L. Sun, W. Y. Han, X. Tian and S. Z. Zhong, Life cycle based emergy analysis on China's cement production, J. Clean. Prod., 131 (2016), 1-8. [6] W. D. Cook and L. M. Seiford, Data envelopment analysis (DEA)-Thirty years on, Eur. J. Oper. Res., 192 (2009), 1-17.  doi: 10.1016/j.ejor.2008.01.032. [7] R. F$\ddot{a}$re and S. Grosskopf, Network DEA, Socio. Econ. Plann. Sci., 4 (2000), 35-49. [8] C. Kao and S. T. Liu, Fuzzy efficiency measures in data envelopment analysis, Fuzzy Sets Syst., 113 (2000), 427-437. [9] H. Mikulcic, H. Cabezas, M. Vujanovic and N. Duic, Environmental assessment of different cement manufacturing processes based on emergy and ecological footprint analysis, J. Clean. Prod., 130 (2016), 1-25. [10] D. Pearce and R. K. Turner, Economics of Natural Resources and the Environment, The Johns Hopkins University Press, Baltimore, 1998. [11] B. Simon, What are the most significant aspects of supporting the circular economy in the plastic industry?, Resources, Conserv. Recy., 141 (2019), 299-300. [12] L. Sun, H. Li, L. Dong, K. Fang, J.Z. Ren, Y. Geng, M. Fujii, W. Zhang, N. Zhang and Z. Liu, Eco-benefits assessment on urban industrial symbiosis based on material flows analysis and emergy evaluation approach: a case of Liuzhou city, China. Resources, Conserv. Recy., 119 (2017), 78-88. [13] H. Wu, Y. Liu, Q. Xia and W. Zhu, Measuring efficiency of recycling systems based on data envelopment analysis (DEA) network: A case from Chinese provincial circular economy, Environ. Eng. Manag. J., 13 (2014), 1089-1099. [14] L. A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets Syst., 1 (1978), 3-28.  doi: 10.1016/0165-0114(78)90029-5. [15] H. Z. Zimmermann, Fuzzy Set Theory and Its Applications, Kluwer-Nijhoff, Boston, 1996. doi: 10.1007/978-94-015-7153-1.
A recycling production system
Notations of a recycling production system
Data set for assessing the recycling production system of EU countries
 DMU $X^1_1$ $X^1_2$ $X^1_3$ $Y^{1g}$ $Y^{1b}$ $X^2$ $Y^{2b}_1$ $Y^{2b}_2$ $Y^{2g}_1$ $Y^{2g}_2$ $Y^{2g}_3$ BE $5004$ $109635$ $36332$ $46000$ $63150004$ $2246$ $401821505$ $260944420$ $4855292916$ $0$ $796941524$ BG $4025$ $9889$ $9662$ $20900$ $120510053$ $262$ $11377402360$ $1228940$ $624743431$ $0$ $47630543$ CZ $5387$ $48753$ $24880$ $34200$ $25380325$ $489$ $421402923$ $8969823$ $1256506220$ $736788139$ $114365359$ DK $3024$ $62642$ $14449$ $49300$ $20981701$ $73$ $611036025$ $453589$ $1077685765$ $0$ $408994682$ DE $43294$ $696913$ $216447$ $49800$ $400049839$ $5139$ $7258425559$ $474079860$ $17091404085$ $10641085528$ $4539988852$ EE $694$ $5212$ $2818$ $30200$ $24277641$ $39$ $1571046418$ $7207$ $523501743$ $271321686$ $61887076$ IE $2248$ $97003$ $11609$ $69100$ $15483387$ $11$ $595176663$ $3974931$ $163521509$ $712096348$ $73569252$ EL $4906$ $22630$ $16702$ $27400$ $69009312$ $1045$ $6101405554$ $2254867$ $223834033$ $559291350$ $14145383$ ES $23016$ $247385$ $82497$ $37200$ $128946782$ $5561$ $6917730969$ $1630152$ $4782727890$ $729903060$ $462686172$ FR $30319$ $539810$ $147158$ $43200$ $323467567$ $10927(p)$ $8912319950$ $529931991$ $17805492324$ $3347460780$ $1751551681$ HR $1833$ $10352$ $6639$ $23800$ $5280553$ $3$ $252644927$ $8025$ $248979558$ $21330946$ $5091818$ IT $25584$ $318657$ $115930$ $37600$ $163997388$ $10323$ $2330759325$ $448068861$ $12940757588$ $16749969$ $663403006$ CY $611$ $3610$ $1758$ $36100$ $3384016$ $37$ $195647721$ $0$ $35308927$ $94608345$ $12836602$ LV $1008$ $5019$ $3820$ $26200$ $2531726$ $86$ $51514453$ $27299$ $181524359$ $2877570$ $17228886$ LT $1481$ $8090$ $5108$ $30700$ $6645689$ $146$ $376219111$ $265115$ $222175413$ $27546012$ $38363239$ LU $281$ $10079$ $4038$ $105400$ $10129884$ $136$ $394682009$ $447$ $352026252$ $244913566$ $21366130$ HU $4686$ $24195$ $17865$ $28300$ $15908526$ $159$ $544534135$ $9173793$ $861341728$ $58137346$ $117665638$ MT $212$ $2613$ $583$ $40100$ $1971253$ $58$ $33840198$ $729659$ $37642029$ $124913458$ $0$ NL $9050$ $154915$ $49517$ $52800$ $141027997$ $3827$ $6484656382$ $123312495$ $6429380745$ $0$ $1065450090$ AT $4535$ $90187$ $28127$ $49000$ $61226583$ $174$ $2809451363$ $\tilde{Y}^{2b}_2$ $2264108063$ $675844441$ $\tilde{Y}^{2g}_3$ PL $18393$ $84961$ $66652$ $28300$ $181990641$ $372$ $5095006125$ $63893549$ $8399307294$ $4036605469$ $604251699$ PT $5207$ $31296$ $16114$ $29600$ $14734417$ $425.5(e)$ $510697523$ $3546142$ $640596720$ $140172510$ $178428819$ RO $8939$ $43109$ $22280$ $22900$ $177557398$ $508$ $16702885666$ $10101348$ $714524681$ $77559190$ $250668909$ SI $996$ $7859$ $4875$ $32900$ $5517787$ $42$ $38252271$ $4285174$ $332376636$ $150325015$ $26539590$ SK $2762$ $19021$ $10418$ $32000$ $10606352$ $319.7(p)$ $506588801$ $4865030$ $424682144$ $49836276$ $74662989$ FI $2687$ $51490$ $25248$ $43400$ $122869413$ $75$ $10816368847$ $5368598$ $908084245$ $0$ $557119580$ SE $5245$ $123749$ $32590$ $50800$ $141622198$ $760$ $10806010637$ $22813777$ $1702475279$ $694274083$ $936645999$ UK $33693$ $437140$ $133688$ $43800$ $277272474$ $13601$ $10410809602$ $735540574$ $13457214835$ $2175093717$ $948588194$ 1. BE: Belgium; BG: Bulgaria; CZ: Czechia; DK: Denmark; DE: Germany; EE: Estonia; IE: Ireland; EL: Greece; ES: Spain; FR: France; HR: Croatia; IT: Italy; CY: Cyprus; LV: Latvia; LT: Lithuania; LU: Luxembourg; HU: Hungary; MT: Malta; NL: Netherlands; AT: Austria; PL: Poland; PT: Portugal; RO: Romania; SI: Slovenia; SK: Slovakia; FI: Finland; SE: Sweden; UK: United Kingdom2. Available flags: p: provisional; e: estimated
 DMU $X^1_1$ $X^1_2$ $X^1_3$ $Y^{1g}$ $Y^{1b}$ $X^2$ $Y^{2b}_1$ $Y^{2b}_2$ $Y^{2g}_1$ $Y^{2g}_2$ $Y^{2g}_3$ BE $5004$ $109635$ $36332$ $46000$ $63150004$ $2246$ $401821505$ $260944420$ $4855292916$ $0$ $796941524$ BG $4025$ $9889$ $9662$ $20900$ $120510053$ $262$ $11377402360$ $1228940$ $624743431$ $0$ $47630543$ CZ $5387$ $48753$ $24880$ $34200$ $25380325$ $489$ $421402923$ $8969823$ $1256506220$ $736788139$ $114365359$ DK $3024$ $62642$ $14449$ $49300$ $20981701$ $73$ $611036025$ $453589$ $1077685765$ $0$ $408994682$ DE $43294$ $696913$ $216447$ $49800$ $400049839$ $5139$ $7258425559$ $474079860$ $17091404085$ $10641085528$ $4539988852$ EE $694$ $5212$ $2818$ $30200$ $24277641$ $39$ $1571046418$ $7207$ $523501743$ $271321686$ $61887076$ IE $2248$ $97003$ $11609$ $69100$ $15483387$ $11$ $595176663$ $3974931$ $163521509$ $712096348$ $73569252$ EL $4906$ $22630$ $16702$ $27400$ $69009312$ $1045$ $6101405554$ $2254867$ $223834033$ $559291350$ $14145383$ ES $23016$ $247385$ $82497$ $37200$ $128946782$ $5561$ $6917730969$ $1630152$ $4782727890$ $729903060$ $462686172$ FR $30319$ $539810$ $147158$ $43200$ $323467567$ $10927(p)$ $8912319950$ $529931991$ $17805492324$ $3347460780$ $1751551681$ HR $1833$ $10352$ $6639$ $23800$ $5280553$ $3$ $252644927$ $8025$ $248979558$ $21330946$ $5091818$ IT $25584$ $318657$ $115930$ $37600$ $163997388$ $10323$ $2330759325$ $448068861$ $12940757588$ $16749969$ $663403006$ CY $611$ $3610$ $1758$ $36100$ $3384016$ $37$ $195647721$ $0$ $35308927$ $94608345$ $12836602$ LV $1008$ $5019$ $3820$ $26200$ $2531726$ $86$ $51514453$ $27299$ $181524359$ $2877570$ $17228886$ LT $1481$ $8090$ $5108$ $30700$ $6645689$ $146$ $376219111$ $265115$ $222175413$ $27546012$ $38363239$ LU $281$ $10079$ $4038$ $105400$ $10129884$ $136$ $394682009$ $447$ $352026252$ $244913566$ $21366130$ HU $4686$ $24195$ $17865$ $28300$ $15908526$ $159$ $544534135$ $9173793$ $861341728$ $58137346$ $117665638$ MT $212$ $2613$ $583$ $40100$ $1971253$ $58$ $33840198$ $729659$ $37642029$ $124913458$ $0$ NL $9050$ $154915$ $49517$ $52800$ $141027997$ $3827$ $6484656382$ $123312495$ $6429380745$ $0$ $1065450090$ AT $4535$ $90187$ $28127$ $49000$ $61226583$ $174$ $2809451363$ $\tilde{Y}^{2b}_2$ $2264108063$ $675844441$ $\tilde{Y}^{2g}_3$ PL $18393$ $84961$ $66652$ $28300$ $181990641$ $372$ $5095006125$ $63893549$ $8399307294$ $4036605469$ $604251699$ PT $5207$ $31296$ $16114$ $29600$ $14734417$ $425.5(e)$ $510697523$ $3546142$ $640596720$ $140172510$ $178428819$ RO $8939$ $43109$ $22280$ $22900$ $177557398$ $508$ $16702885666$ $10101348$ $714524681$ $77559190$ $250668909$ SI $996$ $7859$ $4875$ $32900$ $5517787$ $42$ $38252271$ $4285174$ $332376636$ $150325015$ $26539590$ SK $2762$ $19021$ $10418$ $32000$ $10606352$ $319.7(p)$ $506588801$ $4865030$ $424682144$ $49836276$ $74662989$ FI $2687$ $51490$ $25248$ $43400$ $122869413$ $75$ $10816368847$ $5368598$ $908084245$ $0$ $557119580$ SE $5245$ $123749$ $32590$ $50800$ $141622198$ $760$ $10806010637$ $22813777$ $1702475279$ $694274083$ $936645999$ UK $33693$ $437140$ $133688$ $43800$ $277272474$ $13601$ $10410809602$ $735540574$ $13457214835$ $2175093717$ $948588194$ 1. BE: Belgium; BG: Bulgaria; CZ: Czechia; DK: Denmark; DE: Germany; EE: Estonia; IE: Ireland; EL: Greece; ES: Spain; FR: France; HR: Croatia; IT: Italy; CY: Cyprus; LV: Latvia; LT: Lithuania; LU: Luxembourg; HU: Hungary; MT: Malta; NL: Netherlands; AT: Austria; PL: Poland; PT: Portugal; RO: Romania; SI: Slovenia; SK: Slovakia; FI: Finland; SE: Sweden; UK: United Kingdom2. Available flags: p: provisional; e: estimated
$\alpha$-cuts of the fuzzy system efficiencies
 DMU System Sub-system 1 Sub-system 2 Belgium (0.4029, 0.4185) (0.0014, 0.0018) (0.9593, 1.0000) Bulgaria (0.1067, 0.1376) (0.1077, 0.1377) (0.0272, 0.0309) Czechia (0.3176, 0.3177) (0.0678, 0.0734) (0.9328, 0.9433) Denmark (0.4708, 0.5354) (0.0907, 0.1084) (1.0000, 1.0000) Germany (0.2082, 0.2963) (0.0024, 0.0032) (0.5834, 0.6320) Estonia (0.6148, 0.6150) (0.2746, 0.2897) (0.9908, 1.0000) Ireland (0.6397, 0.6433) (0.3472, 0.3494) (1.0000, 1.0000) Greece (0.0768, 0.0788) (0.0756, 0.0789) (0.0025, 0.0031) Spain (0.0159, 0.0252) (0.0285, 0.0433) (0.0155, 0.0243) France (0.0823, 0.0839) (0.0006, 0.0006) (0.1034, 0.1038) Croatia (1.0000, 1.0000) (1.0000, 1.0000) (1.0000, 1.0000) Italy (0.3056, 0.3398) (0.0079, 0.0085) (0.7341, 0.8466) Cyprus (0.9489, 1.0000) (1.0000, 1.0000) (0.9942, 1.0000) Latvia (1.0000, 1.0000) (1.0000, 1.0000) (1.0000, 1.0000) Lithuania (0.2470, 0.2473) (0.2472, 0.2473) (0.0243, 0.0720) Luxembourg (1.0000, 1.0000) (1.0000, 1.0000) (1.0000, 1.0000) Hungary (0.1631, 0.1789) (0.0167, 0.0277) (0.2482, 0.2539) Malta (0.9553, 1.0000) (0.9228, 1.0000) (0.9768, 1.0000) Netherlands (0.0257, 0.0544) (0.0184, 0.0407) (0.0488, 0.0603) Austria (0.1721, 0.1732) (0.0723, 0.0746) (0.1936, 0.1965) Poland (0.4481, 0.4577) (0.0058, 0.0063) (1.0000, 1.0000) Portugal (0.0616, 0.0839) (0.0038, 0.0047) (0.1947, 0.1913) Romania (0.0346, 0.0680) (0.0043, 0.0085) (0.0689, 0.0732) Slovenia (0.6530, 0.6577) (0.3634, 0.3735) (1.0000, 1.0000) Slovakia (0.1095, 0.1096) (0.1096, 0.1096) (0.0637, 0.0639) Finland (0.3443, 0.5039) (0.0065, 0.0079) (0.8635, 1.0000) Sweden (0.1767, 0.1825) (0.0527, 0.0636) (0.1865, 0.2012) United Kingdom (0.0291, 0.0298) (0.0022, 0.0029) (0.0320, 0.0321)
 DMU System Sub-system 1 Sub-system 2 Belgium (0.4029, 0.4185) (0.0014, 0.0018) (0.9593, 1.0000) Bulgaria (0.1067, 0.1376) (0.1077, 0.1377) (0.0272, 0.0309) Czechia (0.3176, 0.3177) (0.0678, 0.0734) (0.9328, 0.9433) Denmark (0.4708, 0.5354) (0.0907, 0.1084) (1.0000, 1.0000) Germany (0.2082, 0.2963) (0.0024, 0.0032) (0.5834, 0.6320) Estonia (0.6148, 0.6150) (0.2746, 0.2897) (0.9908, 1.0000) Ireland (0.6397, 0.6433) (0.3472, 0.3494) (1.0000, 1.0000) Greece (0.0768, 0.0788) (0.0756, 0.0789) (0.0025, 0.0031) Spain (0.0159, 0.0252) (0.0285, 0.0433) (0.0155, 0.0243) France (0.0823, 0.0839) (0.0006, 0.0006) (0.1034, 0.1038) Croatia (1.0000, 1.0000) (1.0000, 1.0000) (1.0000, 1.0000) Italy (0.3056, 0.3398) (0.0079, 0.0085) (0.7341, 0.8466) Cyprus (0.9489, 1.0000) (1.0000, 1.0000) (0.9942, 1.0000) Latvia (1.0000, 1.0000) (1.0000, 1.0000) (1.0000, 1.0000) Lithuania (0.2470, 0.2473) (0.2472, 0.2473) (0.0243, 0.0720) Luxembourg (1.0000, 1.0000) (1.0000, 1.0000) (1.0000, 1.0000) Hungary (0.1631, 0.1789) (0.0167, 0.0277) (0.2482, 0.2539) Malta (0.9553, 1.0000) (0.9228, 1.0000) (0.9768, 1.0000) Netherlands (0.0257, 0.0544) (0.0184, 0.0407) (0.0488, 0.0603) Austria (0.1721, 0.1732) (0.0723, 0.0746) (0.1936, 0.1965) Poland (0.4481, 0.4577) (0.0058, 0.0063) (1.0000, 1.0000) Portugal (0.0616, 0.0839) (0.0038, 0.0047) (0.1947, 0.1913) Romania (0.0346, 0.0680) (0.0043, 0.0085) (0.0689, 0.0732) Slovenia (0.6530, 0.6577) (0.3634, 0.3735) (1.0000, 1.0000) Slovakia (0.1095, 0.1096) (0.1096, 0.1096) (0.0637, 0.0639) Finland (0.3443, 0.5039) (0.0065, 0.0079) (0.8635, 1.0000) Sweden (0.1767, 0.1825) (0.0527, 0.0636) (0.1865, 0.2012) United Kingdom (0.0291, 0.0298) (0.0022, 0.0029) (0.0320, 0.0321)

Impact Factor: