Special Cases in Linear Programming H. R. Alvarez A., Ph. D. 1
Data Envelopment Analysis
Objective To compare technical efficiency of different Decision Making Units (DMU) The comparison is made as a function of using inputs optimally, creating an efficient ideal unit. H. R. Alvarez A., Ph. D. 3
Paretto Koopman Efficiency A Decision Making Unit (DMU) is not efficient in producing its outputs (from a given amount of inputs) if it can be shown that some distribution of resources will result in the same amount of this output with less of the same resources and no more of any other resources. Conversely, a firm is efficient if this is not possible H. R. Alvarez A., Ph. D. 4
An efficient production function According to Farrell, the production function: Y 0 = Y(y 1, y 2,, y m )= (x 1, x 2., x k ) Is efficient is any other vector Y i produces the same elements such that Y i Y 0 y, x Y i is known as an attainable point of the efficient production function Y 0 If this is true, Y i is technically possible since it produces at most the same amount of the ideal function Y 0 H. R. Alvarez A., Ph. D. 5
Characteristics of an efficient function Convexity: The function has a negative slope that creates an envelope between (0, ); (, 0) this condition guarantees that if two points are attainable in practice, then so is any point representing a weighted average of it. Constant returns to scale: an increase (decrease) in the inputs will produce a proportional increase (decrease) in the outputs. H. R. Alvarez A., Ph. D. 6
y P P x H. R. Alvarez A., Ph. D. 7
Example Three decision making units (DMUs) use two inputs x 1 y x 2 to produce one product y such that: DMU y x 1 x 2 1 15 6 2 2 12 4 5 3 20 10 8 H. R. Alvarez A., Ph. D. 8
Normalizing the input levels DMU x 1 /y x 2 /y 1 6/15 = 0.400 2/15 = 0.133 2 4/12 = 0.333 5/12 = 0.417 3 10/20 = 0.500 8/20 = 0.400 H. R. Alvarez A., Ph. D. 9
Efficient frontier Ideal efficient unit DMU 3 DMU 3' Efficiency of DMU 3 DMU3 H. R. Alvarez A., Ph. D. 10
DEA formulation Deveolped by Charnes, Cooper andrhodes (1984) Non parametric approach based on fractional programming Does not require a predefined function H. R. Alvarez A., Ph. D. 11
max h s.t. : o s r1 m i1 u r i y v x r, 0 i, 0 s r1 m i1 u r i y v x r,j i,j 1 j 1,2,..., n;r 1,2,..., s;i 1,2,..., m y r,j, x i,j,u r,v i 0 H. R. Alvarez A., Ph. D. 12
Where: y r,j : Is the r-th output of the j-th DMU x i,j : Is the i-th input of the j-th DMU u r,j : v i,j : Is the weight of the r-th output in the production function of the j-th DMU Is the weight of the i-th input in the production function of the j-th DMU j=0 : Is the reference unit H. R. Alvarez A., Ph. D. 13
In the formulation Values of x i,j and y r,j are past observations Values of u r,j and v i,j are the decision variables H. R. Alvarez A., Ph. D. 14
Formulation as LP maxh s.t. : 0 s r1 u r y r, 0 m i1 v x i i, 0 1 s r1 u u,v r r i y r,j m i1 0 r, j v x i i,j 0 j 1,2,..., n H. R. Alvarez A., Ph. D. 15
With a dual such that: minw s.t. : n j1 p 0,j 0 y r,j q 0 y r, 0 q 0 x i, 0 n j1 p 0,j x r,j 0 p 0,j 0; q 0 not restrictedin sign H. R. Alvarez A., Ph. D. 16
Input orientation A DMU is not efficient if it is possible to maintain the outputs of the unit at a constant level, or increase them, while decreasing any input without augmenting any other input. Considering the dual formulation, p 0,j will be positive if the corresponding primal constraint defines the DMU as efficient. The set of DMUs that contains all the positive p 0,j is the reference set for DMU 0 H. R. Alvarez A., Ph. D. 17
Determining the new efficient unit x E,i n j1 p 0,j x i,j ; for i 1,2,..., m y E,r n j1 p 0,j y r,j ; for r 1,2,..., s H. R. Alvarez A., Ph. D. 18
Example For the previous case: DMU y x 1 X 2 1 15 6 2 2 12 4 5 3 20 10 8 H. R. Alvarez A., Ph. D. 19
Formulation and solution with QSB
DMU 1 Max. 15u 1 s.t.: 6v 1 + 2v 3 = 1 15u 1-6v 1-2v 3 0 12u 1-4v 1-5v 3 0 20u 1-10v 1-8v 3 0 H. R. Alvarez A., Ph. D. 21
DMU 2 Max. 12u 1 s.t.: 4v 1 + 5v 3 = 1 15u 1-6v 1-2v 3 0 12u 1-4v 1-5v 3 0 20u 1-10v 1-8v 3 0 H. R. Alvarez A., Ph. D. 22
DMU 3 Max. 20u 1 s.t.: 10v 1 + 8v 3 = 1 15u 1-6v 1-2v 3 0 12u 1-4v 1-5v 3 0 20u 1-10v 1-8v 3 0 H. R. Alvarez A., Ph. D. 23
DMU1 Relative efficiency p o,j H. R. Alvarez A., Ph. D. 24
DMU2 Relative efficiency p o,j H. R. Alvarez A., Ph. D. 25
DMU3 Relative efficiency p o,j H. R. Alvarez A., Ph. D. 26
Unit Efficiency DMU 1 100.00 % DMU 2 100.00 % DMU 3 72.61 % H. R. Alvarez A., Ph. D. 27
Efficient unit X E,1 = 0.5941*6 + 0.9241*4 = 7.2610 X E,2 = 0.5941*2 + 0.9241*5 = 5.8087 y E,1 = 0.5941*15 + 0.9241*12=20.0007 DMU y x 1 X 2 1 15 6 2 2 12 4 5 3 20 10 8 H. R. Alvarez A., Ph. D. 28
Validating the efficient DMU H. R. Alvarez A., Ph. D. 29
Output orientation Developed by Bessent and Bessent (1988) This approach considers the difficulties in allocating resources in a competitive market A DMU is not efficient if it is possible to augment any output without increasing any input, and without decreasing any other output This approach presents a formulation slightly different from the dual H. R. Alvarez A., Ph. D. 30
H. R. Alvarez A., Ph. D. 31 General formulation 0 0 1 1 0 0 r r j r, j i,j i,0 r j n j i,j r j n j r,j r, 0 S, S,, y, x m 1,2,..., i for ; x S x s 1,2,..., r for ; S y z y s.t. : max z
Where: z 0 : The efficiency will be given by h 0 =1/z 0 j : S r+, S r - Is the weight for DMU j. Is the decision variable of the problem. : Are slack variables of the constraints H. R. Alvarez A., Ph. D. 32
Defining the efficient unit y E,r z 0 y 0,r S r x E,r x 0,i S r H. R. Alvarez A., Ph. D. 33
Example For the previous case: DMU y x 1 X 2 1 15 6 2 2 12 4 5 3 20 10 8 H. R. Alvarez A., Ph. D. 34
General formulation and solution in WinQSB
DMU 1 DMU y x 1 X 2 1 15 6 2 Max z s.t.: 15z 15d 1 12d 2-20d 3 0 6d 1 + 4d 2 +10d 3 6 2d 1 + 5d 2 + 8d 3 2 2 12 4 5 3 20 10 8 H. R. Alvarez A., Ph. D. 36
DMU 2 DMU y x 1 X 2 1 15 6 2 Max z s.t.: 12z 15d 1 12d 2-20d 3 0 6d 1 + 4d 2 + 10d 3 4 2d 1 + 5d 2 + 8d 3 5 2 12 4 5 3 20 10 8 H. R. Alvarez A., Ph. D. 37
DMU 3 DMU y x 1 X 2 1 15 6 2 2 12 4 5 Max z s.t.: 20z 15d 1 12d 2-20d 3 0 6d 1 + 4d 2 +10d 3 10 2d 1 + 5d 2 + 8d 3 8 3 20 10 8 H. R. Alvarez A., Ph. D. 38
DMU 1 H. R. Alvarez A., Ph. D. 39
DMU 2 H. R. Alvarez A., Ph. D. 40
DMU 3 H. R. Alvarez A., Ph. D. 41
Efficient unit Efficiency of DMU 3 = 1/1.3773 = 0.7261 y E,1 = 1.3773*20 + 0 = 27.546 x E,1 =10 0 = 10 x E,2 = 8 0 = 8 H. R. Alvarez A., Ph. D. 42
DMU 3 Eficiente H. R. Alvarez A., Ph. D. 43
Find the efficienciy of the DMUs shown in the table: H. R. Alvarez A., Ph. D. 44