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In this chapter we described the CCR model in some detail in both its input-oriented and output-oriented versions. 1. We also relaxed assumptions of a positive data set to semipositivity. 2. We defined the production possibility set based on the constant returns-to-scale assumption and developed the CCR model under this assumption. 3. The dual problem of the original CCR model was introduced as ( DLP _o) in (3.6)–(3.9) and the existence of input excesses and output shortfalls clarified by solving this model. (To avoid confusion and to align our terminology with the DEA literature, we referred to the dual as the “envelopment model” and the primal ( LP _o) introduced in Chapter 2 as the “multiplier model.”) 4. In Definition 3.2 we identified a DMU as CCR-efficient if and only if it is ( i ) radial-efficient and ( ii ) has zero-slack in the sense of Definition 3.1. Hence a DMU is CCR-efficient if and only if it has no input excesses and no output shortfalls. 5. Improvement of inefficient DMUs was discussed and formulae were given for effecting the improvements needed to achieve full CCR efficiency in the form of the CCR-projections given in (3.22) and (3.23). 6. Detailed computational procedures for the CCR model were presented in Section 3.6 and an optimal multiplier values ( v, u ) were obtained as the simplex multipliers for an optimal tableau obtained from the simplex method of linear programming.

In this chapter, we introduced the assurance region and cone-ratio methods for combining subjective and expert evaluations with the more objective methods of DEA. 1. Usually expressed in the form of lower and upper bounds, the assurance region method puts constraints on the ratio of input (output) weights or multiplier values. This helps to get rid of zero weights which frequently appear in solution to DEA models. The thus evaluated efficiency score generally drops from its initial (unconstrained) value. Careful choice of the lower and upper bounds is recommended. 2. Not covered in this chapter is the topic of “linked constraints” in which conditions on input and output multipliers are linked. See Problem 6.3. 3. The cone-ratio method confines the feasible region of virtual multipliers v, u, to a convex cone generated by admissible directions. Formulated as a “cone ratio envelopment” this method can be regarded as a generalization of the assurance region approach. 4. Example applications were used to illustrate uses of both of the “assurance region” and “cone ratio envelopment” approaches.

In this chapter we expanded the ability of DEA to deal with variables that are not under managerial control but nevertheless affect performances in ways that need to be taken into account when effecting evaluations. Non-discretionary and categorical variables represent two of the ways in which conditions beyond managerial control can be taken into account in a DEA analysis. Uses of upper or lower bounds constitute yet another approach and, of course, these approaches can be combined in a variety of ways. Finally, uses of Wilcoxon-Mann-Whitney statistics were introduced for testing results in a nonparametric manner when ranking can be employed. Illustrative examples were supplied along with algorithms that can be used either separately or with the computer code DEA-Solver. We also showed how to extend DEA in order to deal with production possibility sets (there may be more than one) that are not convex. Finally we provided examples to show how new results can be secured when DEA is applied to such sets to test the efficiency of organization forms (and other types of activities) in ways that were not otherwise available.

This chapter introduced the concept of super-efficiency and presented two types of approach for measuring super-efficiency: radial and non-radial. Super-efficiency measures are widely utilized in DEA applications for many purposes, e.g., ranking efficient DMUs, evaluating the Malmquist productivity index and comparing performances of two groups (the bilateral comparisons model in Chapter 7).