Data Warehousing
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Dantzig-Wolfe Decomposition Algorithm are regarded as responsible for converting inputs into A variant of the simplex method designed to solve block-angular linear programs in which the blocks define subproblems. The problem is transformed into one that finds a solution in terms of convex combinations of the extreme points of the subproblems.
See ▶ Block-Angular System ▶ Decomposition Algorithms
References
outputs. Examples of its uses have included hospitals and U.S. Air Force Wings, or their subdivisions, such as surgical units and squadrons. The definition of a DMU is generic and flexible. The objective is to identify sources and to estimate amounts of inefficiency in each input and output for every DMU included in a study. Uses that have been accommodated include: (i) discrete periods of production in a plant producing semiconductors in order to identify when inefficiency occurred; and (ii) marketing regions to which advertising and other sales activities have been directed in order to identify where inefficiency occurred. Inputs as well as outputs may be multiple and each may be measured in different units. A variety of models have been developed for implementing the concepts of DEA, for example, the following dual pair of linear programming models:
Dantzig, G. (1963). Linear programming and extensions. Princeton, NJ: Princeton University Press. Dantzig, G., & Thapa, M. (2003). Linear programming 2: Theory and extensions. New York: Springer. Dantzig, G., & Wolfe, P. (1960). Decomposition principle for linear programs. Operations Research, 8(1), 101–111.
min h0 ¼ y0 e
m X
subject to 0 ¼ y0 xi0
! sþ r
r¼1 n X
xij lj s i
n X
(1a)
yrj lj sþ r
j¼1 0 lj ; sþ r ; si
and s X
max y0 ¼
DEA (Data Envelopment Analysis) is a data oriented approach for evaluating the performance of a collection of entities called DMUs (Decision Making Units) which
s X
j¼1
Data Envelopment Analysis
Introduction
þ
i¼1
yr0 ¼
William W. Cooper The University of Texas at Austin, Austin, TX, USA
s i
subject to 1 ¼ 0
S.I. Gass, M.C. Fu (eds.), Encyclopedia of Operations Research and Management Science, DOI 10.1007/978-1-4419-1153-7, # Springer Science+Business Media New York 2013
s X r¼1
mr yr0
r¼1 m X
vi xi0
i¼1 m X
mr yrj
vi xij
i¼1
e mr ; vi
(1b)
D
350
where xij ¼ observed amount of input i used by DMUj and yrj ¼ observed amount of output r produced by DMUj, with i ¼ 1, . . ., m; r ¼ 1, . . ., s; j ¼ 1, . . ., n. All inputs and outputs are assumed to be positive. (This condition may be relaxed (Charnes et al. 1991).
Efficiency The orientation of linear programming has changed here from ex-ante uses, for planning, and apply it to choices already made ex-post, for purposes of evaluation and control. To evaluate the performance of any DMU, (1) is applied to the input–output data for all DMUs in order to evaluate the performance of each DMU in accordance with the following definition: Efficiency — Extended Pareto-Koopmans Definition : Full (100%) efficiency is attained by any DMU if and
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