By Subhash C. Ray

This ebook describes the tactic of knowledge envelopment research that makes use of mathematical programming innovations to acquire measures of potency of person varieties from their saw enter and output amounts. the strategy allows developing reasonable input-output goals for the firm's managers. a company 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. dimension of technical potency is critical for functionality evaluate and gives an aim foundation for differential rewards within the context of creation.

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

**Example text**

Proof. Let DO (x, y) = δ. That means that δ is the smallest positive real number such that (x, 1δ y) ∈ T. Now define yˆ = αy. Let DO (x, yˆ ) = β. This means that, for a given α, β is the smallest real number such that (x, βα y) ∈ T. We need to show that β = αδ. Suppose that this is not true and β < αδ. That is, βα < δ. But in that case, DO (x, y) cannot be δ because there exists another real number γ = βα smaller than δ such that (x, γ1 y) ∈ T. Alternatively, assume that β > αδ. But, because the input–output pair (x, 1δ y) is feasible, so is the input–output 44 Appendix to Chapter 2 1 yˆ ).

Note that when DO (x, y) is greater than unity, the output bundle y cannot be produced from the input bundle x. Only some proportionately scaled-down output bundle will be feasible. On the other hand, if DO (x, y) is less than unity, then a proportionately expanded output bundle will be feasible. Hence, by free disposability of outputs, the bundle y is also feasible. Thus, an alternative specification of the production possibility set is T = {(x, y) : DO (x, y) ≤ 1}. 2) Consider the following 2-input, 2-output example.

J=1 N µ j y2 j ≥ φy2t ; j=1 N µ j x1 j < x1t ; j=1 N µ j x2 j ≤ x2t ; j=1 φ free; µ j ≥ 0; ( j = 1, 2, . . , N ). 29) Thus, clearly φ1∗ from this problem equals θ ∗ from the previous problem. Further, by standard duality results, θ ∗ equals p t∗ y t . 1. The hypothetical input and output quantities for six firms. 5 Data Envelopment Analysis 33 To evaluate the technical efficiency of firm C, we solve the following LP problem: max φ s. t. 30) 3λ A + 5λ B + 7λC + 8λ D + 4λ E + 6λ F ≤ 7; λ A , λ B , .