3. The basic model of the DEA with
constant returns to scale I
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3.1
Benchmarking and productivity form: The idea
3.2
The maximization problem
3.3
The transformation of the maximization problem
3.4
The solution to the maximization problem
3.5
The result to the maximization problem
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3.1 The idea (I)
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DEA - Data Envelopment Analysis – Non-Parametric Frontier-Analysis
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CCR-Model: Charnes, Cooper und Rhodes (EJOR 1978)
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Production technology
– The production technology is characterized by constant returns to scale
(CRS).
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Classical approach:
– In the first step one determines the total factor productivty for every firm; for
example with an traditional index for the total factor productivity.
– In the second step the productivity index of the various firms will be put into
the right perspective by dividing the index of every firm with the firm that has
the highest level of productivity given their production technology (relative
productivity).
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3.1 The idea (II)
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Data Envelopment Analysis
– Measurement of performance and comparison with others (benchmark) will
be conducted simultaneously.
– By this the productivity index of a firm will be maximized under constraints
which ensure that the values of the productivity indices of every single firm
are only in the interval of (0,1].
– The constraints of the maximization problem normalize the efficiency to the
interval of (0,1].
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3.2 The maximization problem (I)
Linear Productivity Index
(1) Firms
(2) Outputs yr
Aggregation weighting factor pr (prices)
(3) Inputs xj
Aggregation weighting factor qj (prices)
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3.2 The maximization problem (II)
Restricted maximization problem
(Matrix and vector notation)
maximize the productivity of firm i
given the productivity of each firm not
larger than 1
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3.3 The transformation of the maximization problem (I)
Charnes-Cooper-Transformation
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Modified maximization problem
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3.3 Example (I)
Productivity index
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3.3 Example (II)
Maximization problem
for firm A
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Charnes-Cooper-Transformation for
firm A
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3.3 Example (III)
Modified maximization problem
for firm A
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3.4 Example I
Efficiency indices and aggregation weighting factors
for firms A, B and C
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3.5 Result of the maximization problem (I)
Efficiency measures
Example
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3.5 Result of the maximization problem(II)
Input aggregation weights absolute:
„production function“
„Marginal productivity“
Input aggregation weights relativ: (only two inputs are varied)
„MRS“
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„Isoquant“
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3.5 Result of the maximization problem (III)
Output aggregation weights absolute:
Output aggregation weights relative: (only two outputs are varied )
„MRT“
Input and output aggregation weights
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3.5 Example I (I)
Example (for i = A,C):
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3.5 Example I (II)
Slope of isoquant
Efficiency of B
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3.5 Example II (I)
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3.5 Example II (II)
Production function
Marginal productivity
Average productivity
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3.5 Example II (III)
Slope:
Straight line b:
Input waste of N:
Efficiency of N:
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Best-practice-inputlevel for the
production of y=3
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