Dantzig-Wolfe Decomposition Algorithm
Dantzig-Wolfe Decomposition Algorithm
William W. Cooper
The University of Texas at Austin, Austin, TX, USA
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 are regarded as responsible for converting inputs into
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 everyDMUincluded in a study. Uses that have been accommodated include: (i) discrete periods of production in a plant producing
semiconductors in order to identifywhen inefficiency occurred; and (ii) marketing regions towhich advertising andother sales activitieshave been directed in order to identify where inefficiency occurred. Inputs as well as outputs may be multiple and eachmay bemeasured 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:
Introduction
DEA (Data Envelopment Analysis) is a data oriented
approach for evaluating the performance of a collection
of entities called DMUs (Decision Making Units) which
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
only if none of its inputs or outputs can be improved
without worsening some of its other inputs or outputs.
This definition has the advantage of avoiding the
need for assigning a priori weights or other measures of
relative importance to any input or output. In most
management or social science applications, the
theoretically possible levels of efficiency will not be
known. For empirical use, the preceding definition is
therefore replaced by the following:
Relative Efficiency: A DMU is to be rated as fully (100%)
efficient if and only if the performances of other DMUs
do not show that some of its inputs or outputs can be
improved without worsening some of its other inputs or
outputs.
To implement this definition, it is necessary only to
designate any DMUj as DMU0 with inputs xi0 and
outputs yr0 and then apply (1) to the input and output
data recorded for the collection of DMUj, j ¼ 1, . . ., n.
Leaving this DMUj ¼ DMU0 in the constraints insures
that solutions will always exist with an optimal
y0 ¼ y0 1. The above definition applied to
(1) then gives
DEA Efficiency: The performance of DMU0 is fully
(100%) efficient if and only if, at an optimum, both (i)
Py0 ¼ 1, and (ii) all slacks ¼ 0 in (1a) or, equivalently, s
r¼1 m r yr0 ¼ 1 in (1b), where ∗ represents an optimal
value.
William W. Cooper
The University of Texas at Austin, Austin, TX, USA
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 are regarded as responsible for converting inputs into
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 everyDMUincluded in a study. Uses that have been accommodated include: (i) discrete periods of production in a plant producing
semiconductors in order to identifywhen inefficiency occurred; and (ii) marketing regions towhich advertising andother sales activitieshave been directed in order to identify where inefficiency occurred. Inputs as well as outputs may be multiple and eachmay bemeasured 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:
Introduction
DEA (Data Envelopment Analysis) is a data oriented
approach for evaluating the performance of a collection
of entities called DMUs (Decision Making Units) which
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
only if none of its inputs or outputs can be improved
without worsening some of its other inputs or outputs.
This definition has the advantage of avoiding the
need for assigning a priori weights or other measures of
relative importance to any input or output. In most
management or social science applications, the
theoretically possible levels of efficiency will not be
known. For empirical use, the preceding definition is
therefore replaced by the following:
Relative Efficiency: A DMU is to be rated as fully (100%)
efficient if and only if the performances of other DMUs
do not show that some of its inputs or outputs can be
improved without worsening some of its other inputs or
outputs.
To implement this definition, it is necessary only to
designate any DMUj as DMU0 with inputs xi0 and
outputs yr0 and then apply (1) to the input and output
data recorded for the collection of DMUj, j ¼ 1, . . ., n.
Leaving this DMUj ¼ DMU0 in the constraints insures
that solutions will always exist with an optimal
y0 ¼ y0 1. The above definition applied to
(1) then gives
DEA Efficiency: The performance of DMU0 is fully
(100%) efficient if and only if, at an optimum, both (i)
Py0 ¼ 1, and (ii) all slacks ¼ 0 in (1a) or, equivalently, s
r¼1 m r yr0 ¼ 1 in (1b), where ∗ represents an optimal
value.
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