DEA cross-efficiency (Sexton, Silkman & Hogan, 1986)

One thing is added; the input/output split, the separate optimisation for every unit, and the "no one can exceed a hundred per cent" constraint do not change.

The evaluation goes beyond a unit's own score. Base DEA (the CCR model) finds, for every unit, the multipliers (weights) that show that unit in the most advantageous light, and uses these multipliers to compute only that unit's OWN score (self-appraisal). This extension does not stop there: every unit's own optimal multipliers are then applied, in turn, to EVERY OTHER unit's data as well. This produces an m×m cross-appraisal matrix for m units; one cell of this matrix shows what score unit j would receive if evaluated using the weights that unit itself chose. Every unit's final cross-efficiency score is the average of the scores it receives in this matrix, taken across every other unit's weights.

Discriminating power increases. In base DEA, no distinction at all is drawn among units on the efficient frontier (theta=1); all are considered equally "efficient." Because cross-efficiency also tests these units against one another's weights, it generally produces a distinction even among units that tied in self-appraisal, and gives a COMPLETE ranking. This is what cross-efficiency chiefly adds to base DEA: an answer to the question of "which unit also looks strong in the eyes of the others," beyond simply "how many units are efficient."

Weight selection is still left self-interested. DecisionMind applies Sexton, Silkman and Hogan's (1986) base form in this extension: the weights every unit uses in the cross-evaluation are chosen only to maximise that unit's OWN score; what these weights do to other units is not optimised. Doyle and Green's (1994) later-proposed "benevolent" (trying to keep other units' scores as high as possible too) or "aggressive" (trying to keep other units' scores as low as possible) secondary-objective forms are not applied in this extension; where more than one optimal weight set exists, which one is chosen is left to the linear-programming solver's own internal rule.

DecisionMind fixes, for this extension, the construction of the cross-evaluation matrix and simple (self-interested, no secondary objective) weight selection.

The final cross-efficiency average lies between 0 and 1, as base DEA's theta does, and higher is better; but its meaning differs. Theta answers the question "how good does this unit look with the most advantageous weights IT ITSELF chose." The cross-efficiency average answers "how good does this unit look on average, with the weights everyone, INCLUDING ITSELF, has chosen."

This difference produces an important consequence: two units that both score theta=1 in self-appraisal can come out very different from one another in cross-efficiency, and either can even fall below a unit that was not efficient in self-appraisal at all. The reason is that a unit scoring theta=1 only means that a narrow weight set best suited to ITS OWN profile could be found; applied to a unit with a different profile, this same weight set can produce a very low score.

Thus instead of writing:

"This branch came out efficient in DEA (θ=1), so it really is the best-managed branch"

the report should read:

"This branch is efficient with the weights it chose itself; the cross-efficiency average also factors in how other branches, with their own weights, see this branch, and this average can either confirm its self-appraised superiority or reveal that superiority to be the product of a one-sided weight choice"

When more than one unit scores theta=1 in self-appraisal, and a distinction, a complete ranking, is also wanted among these units. It is also worth using when it is wanted to report not only how every unit looks with its own weights, but also how it stands "in the eyes of its peers" (situations such as peer assessment, benchmarking reports, or incentive allocation).

If the aim is only to distinguish who is on the efficient frontier from who is not, and a complete ranking is not needed, base DEA is sufficient; cross-efficiency brings extra computational load and, in small datasets (especially single-input or single-output problems), it will not always provide extra discriminating power (see Case 1). Base DEA's exit conditions apply here too: where the aim is to measure efficiency rather than preference, and weights are not to be supplied from outside.

Assuming cross-efficiency will always produce a different order from self-appraisal. In some datasets, especially problems with a single input or few units, all units' optimal weights can converge on the same corner; in that case the cross-efficiency average comes out identical, one for one, to the self-appraisal theta, and provides no extra distinction (see Case 1). This is not a computational error, it stems from the structure of the data; but if unnoticed, it is possible to say "a cross-evaluation was carried out" when in fact no extra information at all was produced.

Trying to feed a weight or direction set visible in the manifest into DEA as an outside "importance weight." Cross-efficiency, like base DEA, does not take weights; every unit's multipliers are derived from its own data through linear programming. Input/output direction (which criterion is an input, which an output) is supplied from outside, but importance weighting is not.

Assuming a benevolent or aggressive secondary-objective form has been applied. This DecisionMind extension does not apply Doyle and Green's (1994) benevolent or aggressive forms when choosing among more than one optimal weight set; it uses only Sexton, Silkman and Hogan's (1986) base, self-interested form. This means that, where more than one optimal weight exists, which one is chosen can depend on the solver.

The governing principle is this:

Cross-efficiency shows how a unit looks not only with its own chosen weights but also with its peers' weights; but whether this extra information genuinely provides a distinction depends on the data, and the report must show that this has been checked.