Transcript 下載/瀏覽

南台科技大學 專題討論報告
指導老師：黃振勝
姓
名：賴佳琪
學
號：M98U0101
中華民國98年12月30日
Report title
Relationship between MOLP and
DEA based on output-orientated
CCR dual model
Author：F. Hosseinzadeh Lotfi, G.R. Jahanshahloo, M.
Soltanifar, A. Ebrahimnejad, S.M. Mansoorzadeh
Source： Expert Systems with Applications, In Press,
Uncorrected Proof, Available online 26 November 2009
Program





Introduction
DEA models
Equivalence between DEA and MOLP
An interactive multi objective programming
Conclusion
2016/7/13
3
Introduction(1/2)


DEA model does not include a decision maker
(DM)’s preference structure or value
judgments while measuring relative efficiency,
with no or minimal input from the DM.
all the above-mentioned techniques would
require prior articulated preference knowledge
from the DM, which in most cases can be
subjective and difficult to obtain.
2016/7/13
4
Introduction(2/2)

Multi objective programming methods such
as multiple objective linear programming
(MOLP) are techniques used to solve such
multiple criteria decision making (MCDM)
problems.
2016/7/13
5
DEA models

DEA is a nonparametric frontier estimation
methodology based on linear programming
for evaluating relative efficiency of a set of
comparable DMU that share common
functional goals.
 Output-orientated
CCR primal model
 Output-orientated CCR dual model
2016/7/13
6
Output-orientated CCR primal model
m
Min
s.t
h0   vi xi 0
i 1
m
s
i 1
s
r 1
 vi xij   ur yij  0 j  1, 2, . .. , n (1)
(1)
 u r yr 0  1
r 1
vi , ur  0 i  1,2,..., m ; r  1,2,..., s
2016/7/13
7
Output-orientated CCR dual model
Max h0   0
n
s.t
  j yr 0   0 yr 0 r  1, 2, . . ., s
j 1
n
  j xij  xi 0 i  1, 2, . .., m
(2)
j 1
 j  0 j  1, 2, . . ., n
2016/7/13
8
Equivalence between
DEA and MOLP(1/7)

Suppose an optimization problem has s objectives
reflecting the different purposes and desires of the
DM.
Max f ( )  [ f1 ( ), f 2 ( ), ..., f r ( ),..., f s ( )]
s.t
 
2016/7/13
(3)
9
Equivalence between
DEA and MOLP(2/7)

In order to reach to a special nondominated extreme
point, the MOLP formulation (3) can be written in
min-ordering approach as follows:
Max Min f r ( )
1r  s
s.t
(4)
 
Max 
s.t
2016/7/13
  f r ( ) r  1, 2, . . . , s
 
(5)
10
Equivalence between
DEA and MOLP(3/7)

From the formulation (2), the output-orientated
CCR dual DEA model can be equivalently rewritten
as follows:
Max  0
n
s.t
  j yrj   0 yr 0 r  1, 2, . . . , s
j 1
n
   0   :   j xij  xi 0 , i  1, 2, . . . , m
(6)
j 1
 j  0, j  1, 2, . . . , n
2016/7/13
11
Equivalence between
DEA and MOLP(4/7)

Suppose for yro any r=1, 2, . . . , s and suppose in
formulation (5) fr () defined as follows:
1 n
t
f r ( ) 
  j yrj r  1, 2, . .. , s   (1,2 ,..., n )
yr 0 j 1
2016/7/13
(7)
12
Equivalence between
DEA and MOLP(5/7)
Max  0
s.t
1 n
0 
  j yrj r  1, 2, . . . , s
yr 0 j 1
n
   0   :   j xij  xi 0 , i  1, 2, . . ., m;
(8)
j 1
 j  0, j  1, 2, . . . , n
2016/7/13
13
Equivalence between
DEA and MOLP(6/7)

Suppose yro > 0 for all r = 1, 2, . . . , s. The outputorientated CCR dual model (6) can be equivalently
transformed to the min-ordering formulation (5)
using formulations (7) and (8) and following
equations:
2016/7/13
  0 ;
(9)
  0
(10)
14
Equivalence between
DEA and MOLP(7/7)
since the formulation (5) gives a special weak
efficient point of formulation (3) (Ehrgott, 2005),
then formulation (6) also gives a special weak
efficient point of following formulation:

1
Max [
y10
n
1
 j y1 j ,

y20
j 1
n
s.t
 x
j 1
j ij
n
1
 j y2 j , . . . ,

ys 0
j 1
 xi 0 i  1, 2, . . . , m
n

j 1
j
ysj ]
(11)
 j  0 j  1, 2, . . . , n
2016/7/13
15
An interactive multi
objective programming(1/2)


The method of Zionts–Wallenius(Z–W) can
be used to design an interactive procedure for
searching for most preferred solution (MPS),
that maximizes the DM’s implicit utility
function.
It is applicable to problem in (3) where the
objective functions are concave and Λ is a
convex set.
2016/7/13
16
An interactive multi
objective programming(2/2)



choose an arbitrary set of positive multipliers
or weights. And generate a composite objective
function or utility function using these
multipliers.
the set of nonbasic variables, a subset of
efficient variables is selected
For each efficient variable a set of trade-offs is
defined by which some objectives are
increased and others reduced.
2016/7/13
17
Conclusion


Establishes an equivalence model between
DEA and MOLP and show how a DEA
problem can be solved interactively by
transforming it into MOLP formulation.
Provides the basis to apply interactive
techniques in MOLP to solve DEA problems
and further locate the MPS along the efficient
frontier for each inefficient DMU.
2016/7/13
18
The end
Thank you for listening