By Subhash C. Ray

This e-book describes the strategy of knowledge envelopment research that makes use of mathematical programming options to procure measures of potency of person varieties from their saw enter and output amounts. the strategy allows constructing lifelike input-output objectives for the firm's managers. an organization is taken into account to be technically inefficient while it fails to yield the utmost volume of output producible from the enter package deal it makes use of. size of technical potency is necessary for functionality review and gives an goal foundation for differential rewards within the context of construction.

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**Extra info for Data Envelopment Analysis: Theory and Techniques for Economics and Operations Research**

**Sample text**

Suppose that r1 and r2 are the prices of the two inputs. 6b) and are the aggregate input quantities for A and B, respectively. 7a) XA r1 x1A + r2 x2A and AP(B) = yB yB = . 7b) But, obviously, the aggregate input bundles represent the input costs of the two firms. Thus, a firm’s average productivity is merely the inverse of its average cost (AC). That is, AP(A) = 1 AC A and AP(B) = 1 . AC B Now suppose that each firm produced two outputs: y1 and y2 . The output prices are q1 and q2 , respectively.

Thus, φ ∗ = min y1t∗ y2t∗ , y1t y2t . 32) Define the output slack variables s1+ = y1t∗ − φ ∗ y1t and s2+ = y2t∗ − φ ∗ y2t . The input slack variables can be similarly defined as s1− = x1t − x1t∗ and s2− = x2t − x2t∗ . It may be recalled that an input–output bundle (x, y) is regarded as Pareto efficient only when (1) it is not possible to increase any output without either reducing some other output or increasing some input, and (2) it is not possible to reduce any input without increasing some other input or reducing some output.

Points PA∗ and PB∗ are vertical projections of the points PA and PB onto the frontier. In both cases, we hold the observed input bundle unchanged and expand the output level till we reach the frontier. This is known as the output-augmenting or output-oriented approach. An alternative is the input-saving or input-oriented approach. 2. In this case, the output level (y A or y B ) remains unchanged and input quantities are reduced proportionately till the frontier is reached. For firm A, the input-oriented projection onto the frontier would be the point PA∗ , where output y A is produced from input x ∗A .