Characteristics and structures of weak efficient surfaces of production possibility sets.

*(English)*Zbl 1253.90124Summary: This paper studies the characteristics and structure of the weak surface of the production possibility set. We apply techniques and methods of transferring a polyhedral cone from its intersection form to its sum form, identify an intersection representation of the production possibility set. We give the structure theorem of weak surface of the production possibility set, which includes three complementary slackness conditions. We define the input weak efficiency and output weak efficiency for different DEA models according to the representation of the intersection form. It investigates the characteristics of the weak surfaces, and proves the structure theorems of input weak DEA efficiency and output weak DEA efficiency. The structure theorems establish weighted combination of inputs and outputs that are weak DEA efficient. Numerical examples are provided for illustration.

##### MSC:

90B50 | Management decision making, including multiple objectives |

90C08 | Special problems of linear programming (transportation, multi-index, data envelopment analysis, etc.) |

90C29 | Multi-objective and goal programming |

##### Software:

Algorithm 457
PDF
BibTeX
XML
Cite

\textit{Q. Wei} et al., J. Math. Anal. Appl. 327, No. 2, 1055--1074 (2007; Zbl 1253.90124)

Full Text:
DOI

##### References:

[1] | Banker, R.D., Estimation most productive scale size using data envelopment analysis, European J. oper. res., 17, 35-44, (1984) · Zbl 0538.90030 |

[2] | Banker, R.D.; Charnes, A.; Cooper, W.W., Some models for estimating technical and scale inefficiencies in data envelopment analysis, Manage. sci., 30, 9, 1078-1092, (1984) · Zbl 0552.90055 |

[3] | Banker, R.D.; Thrall, R.M., Estimation of returns to scale using data envelopment analysis, European J. oper. res., 62, 2, 74-84, (1992) · Zbl 0760.90001 |

[4] | Bron, C.; Kerbosch, J., Algorithm 457—finding all cliques of an undirected graph, Commun. ACM, 16, 575-583, (1973) · Zbl 0261.68018 |

[5] | Charnes, A.; Cooper, W.W.; Rhodes, E., Measuring the efficiency of decision making units, European J. oper. res., 2, 429-444, (1978) · Zbl 0416.90080 |

[6] | Day, D.; Lewin, A.T.; Li, H.; Salazar, R., Strategic leaders in the US brewing industry: A longitudinal analysis of outliers, () |

[7] | Dulá, J.H., Computations in DEA, Pesquisa operacional, 22, 2, 165-182, (2002) |

[8] | Färe, R.; Grosskopf, S., A nonparametric cost approach to scale efficiency, Scand. J. econ., 87, 594-604, (1985) · Zbl 0581.90006 |

[9] | Olesen, O.B.; Petersen, N.C., Identification and use of efficient faces and facets in DEA, J. productiv. anal., 20, 323-360, (2003) · Zbl 0857.90002 |

[10] | Seiford, L.M.; Thrall, R.M., Recent developments in DEA, the mathematical programming approach to frontier analysis, J. econometrics, 46, 7-38, (1990) · Zbl 0716.90015 |

[11] | Wei, Q.L.; Yan, H., Congestion and returns to scale in data envelopment analysis, European J. oper. res., 153, 641-660, (2004) · Zbl 1099.90558 |

[12] | Wei, Q.L.; Yu, G., Analysing the properties of K-cone in generalized data envelopment analysis model, J. econometrics, 80, 63-84, (1997) |

[13] | Yan, H.; Wei, Q.L., A method of transferring cones of intersection-form to cones of sum-form and its applications in DEA models, Internat. J. systems sci., 31, 5, 629-638, (2000) · Zbl 1080.93648 |

[14] | Yan, H.; Wei, Q.L.; Wang, J., Constructing efficient solutions structure of multiobjective linear programming, J. math. anal. appl., 307, 504-523, (2005) · Zbl 1066.90121 |

[15] | Yu, G.; Wei, Q.L.; Brockett, P., A generalized data envelopment analysis model: A unification and extension of existing methods for efficiency analysis of decision making units, Ann. oper. res., 66, 47-89, (1996) · Zbl 0863.90010 |

[16] | Yu, G.; Wei, Q.L.; Brockett, P.; Zhou, L., Construction of all DEA efficient surfaces of the production possibility set under the generalized surfaces of the production possibility set under the generalized DEA model, European J. oper. res., 95, 491-510, (1996) · Zbl 0943.90589 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.