# Efficiency analysis in two-stage structures using fuzzy data envelopment analysis

- 142 Downloads
- 4 Citations

## Abstract

Two-stage data envelopment analysis (TsDEA) models evaluate the performance of a set of production systems in which each system includes two operational stages. Taking into account the internal structures is commonly found in many situations such as seller-buyer supply chain, health care provision and environmental management. Contrary to conventional DEA models as a black-box structure, TsDEA provides further insight into sources of inefficiencies and a more informative basis for performance evaluation. In addition, ignoring the qualitative and imprecise data leads to distorted evaluations, both for the subunits and the system efficiency. We present the fuzzy input and output-oriented TsDEA models to calculate the global and pure technical efficiencies of a system and sub-processes when some data are fuzzy. To this end, we propose a possibilistic programming problem and then convert it into a deterministic interval programming problem using the α-level based method. The proposed method preserves the link between two stages in the sense that the total efficiency of the system is equal to the product of the efficiencies derived from two stages. In addition to the study of technical efficiency, this research includes two further contributions to the ancillary literature; firstly, we minutely discuss the efficiency decompositions to indicate the sources of inefficiency and secondly, we present a method for ranking the efficient units in a fuzzy environment. An empirical illustration is also utilised to show the applicability of the proposed technique.

## Keywords

Data envelopment analysis Efficiency Two-level systems Fuzzy data Ranking## Mathematics Subject Classification

90C05 94D05 90C70 90C90## References

- Agrell PJ, Hatami-Marbini A (2013) Frontier-based performance analysis models for supply chain management: state of the art and research directions. Comput Ind Eng 66(3):567–583CrossRefGoogle Scholar
- Aigner D, Lovell CK, Schmidt P (1977) Formulation and estimation of stochastic frontier production function models. J Econom 6(1):21–37CrossRefGoogle Scholar
- Amirteimoori A (2013) A DEA two-stage decision processes with shared resources. Cent Eur J Operat Res 21(1):141–151CrossRefGoogle Scholar
- Andersen P, Petersen N (1993) A procedure for ranking efficient units in data envelopment analysis. Manage Sci 39:1261–1264CrossRefGoogle Scholar
- Charnes A, Cooper WW, Rhodes E (1978) Measuring the efficiency of decision making units. Eur J Operat Res 2:429–444CrossRefGoogle Scholar
- Cooper WW, Park KS, Yu G (1999) IDEA and AR-IDEA: models for dealing with imprecise data in DEA. Manag Sci 45(4):597–607CrossRefGoogle Scholar
- Emrouznejad A, Tavana M, Hatami-Marbini A (2014) The state of the art in fuzzy data envelopment analysis, in: Performance measurement with fuzzy data envelopment analysis, Studies in fuzziness and soft computing, Springer, vol 309, pp 1–45Google Scholar
- Färe R, Grosskopf S (2000) Network DEA. Soc Econ Plan Sci 34:35–49CrossRefGoogle Scholar
- Färe R, Primont D (1984) Efficiency measures for multiplant firms. Operat Res Lett 3(5):257–260CrossRefGoogle Scholar
- Hatami-Marbini A, Emrouznejad A, Tavana M (2011) A taxonomy and review of the fuzzy data envelopment analysis literature: two decades in the making. Eur J Operat Res 214(3):457–472CrossRefGoogle Scholar
- Hatami-Marbini A, Ebrahimnejad A, Lozano S (2017) Fuzzy efficiency measures in data envelopment analysis using lexicographic multiobjective approach. Comput Ind Eng 105:362–376CrossRefGoogle Scholar
- Kao C (2009) Efficiency measurement for parallel production systems. Eur J Operat Res 196(3):1107–1112CrossRefGoogle Scholar
- Kao C (2012) Efficiency decomposition for parallel production systems. Eur J Operat Res 63(1):64–71CrossRefGoogle Scholar
- Kao C, Hwang SN (2008) Efficiency decomposition in two-stage data envelopment analysis: an application to non-life insurance companies in Taiwan. Eur J Operat Res 185(1):418–429CrossRefGoogle Scholar
- Kao C, Hwang SN (2011) Decomposition of technical and scale efficiencies in two-stage production systems. Eur J Operat Res 211(3):515–519CrossRefGoogle Scholar
- Kao C, Lin PH (2012) Efficiency of parallel production systems with fuzzy data. Fuzzy Sets Syst 198:83–98CrossRefGoogle Scholar
- Kao C, Liu ST (2000) Fuzzy efficiency measures in data envelopment analysis. Fuzzy Sets Syst 113(3):427–437CrossRefGoogle Scholar
- Kao C, Liu ST (2011) Efficiencies of two-stage systems with fuzzy data. Fuzzy Sets Syst 176(1):20–35CrossRefGoogle Scholar
- Land KC, Lovell CA, Thore S (1993) Chance-constrained data envelopment analysis. Manag Decis Econ 14(6):541–554CrossRefGoogle Scholar
- Lertworasirikul S, Fang SC, Joines JA, Nuttle HLW (2003) Fuzzy data envelopment analysis (DEA): a possibility approach. Fuzzy Sets Syst 139(2):379–394CrossRefGoogle Scholar
- Liu ST (2014a) Fuzzy efficiency ranking in fuzzy two-stage data envelopment analysis. Optim Lett 8(2):633–652CrossRefGoogle Scholar
- Liu ST (2014b) Restricting weight flexibility in fuzzy two-stage DEA. Comput Ind Eng 74:149–160CrossRefGoogle Scholar
- Lozano S (2014a) Process efficiency of two-stage systems with fuzzy data. Fuzzy Sets Syst 243:36–49CrossRefGoogle Scholar
- Lozano S (2014b) Computing fuzzy process efficiency in parallel systems. Fuzzy Optim Decis Mak 13(1):73–89CrossRefGoogle Scholar
- Meeusen W, van Den Broeck J (1977) Efficiency estimation from Cobb-Douglas production functions with composed error. Int Econ Rev 435–444CrossRefGoogle Scholar
- Nguyen HT (1978) A note on the extension principle for fuzzy sets. J Math Anal Appl 64(2):369–380CrossRefGoogle Scholar
- Olesen OB, Petersen NC (2016) Stochastic data envelopment analysis—A review. Eur J Oper Res 251(1):2–21CrossRefGoogle Scholar
- Ruiz JL, Sirvent I (2017) Fuzzy cross-efficiency evaluation: a possibility approach. Fuzzy Optim Decis Mak 16(1):111–126CrossRefGoogle Scholar
- Saati SM, Memariani A, Jahanshahloo GR (2002) Efficiency analysis and ranking of DMUs with fuzzy data. Fuzzy Optim Decis Mak 1(3):255–267CrossRefGoogle Scholar
- Seiford LM, Zhu J (1999) Profitability and marketability of the top 55 US commercial banks. Manag Sci 45(9):1270–1288CrossRefGoogle Scholar
- Sengupta JK (1992) A fuzzy systems approach in data envelopment analysis. Comput Math Appl 24(8–9):259–266CrossRefGoogle Scholar
- Wen M, Qin Z, Kang R (2011) Sensitivity and stability analysis in fuzzy data envelopment analysis. Fuzzy Optim Decis Mak 10(1):1–10CrossRefGoogle Scholar
- Yu MM, Chen PC (2011) Measuring air routes performance using a fractional network data envelopment analysis model. Cent Eur J Operat Res 19(1):81–98CrossRefGoogle Scholar
- Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1:3–28CrossRefGoogle Scholar
- Zimmermann HJ (1996) Fuzzy set theory-and its applications, 3rd edn. Kluwer Academic, BostonGoogle Scholar