資料包絡分析法
Data Envelopment Analysis-A Comprehensive
Text with Models,
Applications, References and DEA-Solver
Software
Second Edition
WILLIAM W. COOPER
University of Texas at Austin, U.S.A.
LAWRENCE M. SEIFORD
University of IVIichigan, U.S.A.
KAORU TONE
National Graduate Institute for Policy Studies, Japan
決策品質的比較

效用函數(Utility Functions)
◦ 無法比較，如人飲水，冷暖自知。

單位產出下的成本
◦ 多種產出問題
◦ 多種投入問題
◦ 有些產出或投入無法以金額衡量
多種投入(產出)-以圖書館為例

提供的效果(服務)：
◦ 圖書或設備的使用次數、諮詢次數等

投入成本
◦ 收購書籍、資料庫或設備，人員數量，空
間大小
總要素生產力







總要素生產力(Total Factor Productivity, TFP )
TFP(i)=Yi/Xi i = 1,…n
Yi 代表第i家廠商的產出
Xi 代表第i家廠商的投入
例某行政單位有A,B,C等3個部門
產出(Yi)為某年度i部門的結案公文數
投入(Xi)為某年度i部門的員工數
部門
Y
X
A
60
10
B
80
20
C
90
10
TFP(A)=60/10=6
TFP(B)=80/20=4
TFP(C)=100/10=10
技術效率
技術效率(Technical Efficiency, TE)
 TE(i)=TFP(i)/TFP*
TFP*為所有廠商中最高的TFP(Total Factor
Productivity),本例中以部門C的TFP最高
所以 TE(A)=TFP(A)/TFP(C)=6/10

TE(B)=TFP(B)/TFP(C)=4/10
TE(C)=TFP(C)/TFP(C)=10/10=1
Two Inputs and One Output Case
Store
A
B
C
D
E
F
G
H
I
Employee Xl
4
7
8
4
2
5
6
5.5
6
Floor Area X2
3
3
1
2
4
2
4
2.5
2.5
Sale
1
1
1
1
1
1
1
1
1
Y
production possibility set
To measure inefficiency of A
D and E are called the reference set for A.
P with Input X1= 3.4 and Input X2 = 2.6,
ONE INPUT AND TWO OUTPUTS CASE
Store
A
Employee X
Customers Y1
Sale
Y2
B
1
1
5
C
1
2
7
D
1
3
4
E
1
4
3
F
1
4
6
G
1
5
5
1
6
2
production possibility set
where d(0, D) and d(0, P) mean "distance from zero to D" and "distance
from zero to P," respectively. The ratio is referred to as a "radial
measure” .Because we are concerned with output, however, it is easier
to interpret (1.5) in terms of its reciprocal(1.33). This result means that,
to be efficient, D would have had to increase both of its outputs by 1.33.
This kind of inefficiency which can be eliminated without changing
proportions is referred to as "technical inefficiency."
FIXED AND VARIABLE
WEIGHTS
Hospital A B C D E F G H I
J K L
Doctors 20 19 25 27 22 55 33 31 30 50 53 38
Nurses 151 131 160 168 158 255 235 206 244 268 306 284
Outpatie
100 150 160 180 94 230 220 152 190 250 260 250
nts
Inpatien
90 50 55 72 66 90 88 80 100 100 147 120
ts
(each in units of 100 persons/month)
多投入多產出


人員(X1)與設備(X2)兩種投入
公文(Y1)與專案計畫(Y2)兩種產出
一般的績效評估方式：加權計分（主觀的
給予各投入產出權數）
u1×Y1i+u2×Y2i
TE(i)=
v1×X1i+v2×X2i

u1為Y1的權數，u2為Y2的權數
v1為X1的權數，v2為X2的權數
THE CCR MODEL
the CCR model, which was initially proposed by
Charnes, Cooper and Rhodes in 1978
for each DMU, we formed the virtual input and
output by (yet unknown) weights (vi) and (ur)
FP →LP
Theorem 2.1 The fractional program (FPo) is equivalent to (LPo).
Theorem 2.2 (Units Invariance
Theorem)


The optimal values of max θ = θ * in (2.3) and
(2.7) are independent of the units in which the inputs
and outputs are measured provided these units are
the same for every DMU.
one person can measure outputs in miles and
inputs in gallons of gasoline and quarts of oil
while another measures these same outputs and
inputs in kilometers and liters. They will
nevertheless obtain the same efficiency value
from (2.3) or (2.7)
Definition 2.1 (CCR-Efiiciency)
1. DMUo is CCR-efficient if θ * = 1 and there
exists at least one optimal (v*,u*), with V* > 0 and
u* > 0.
 2. Otherwise, DMUo is CCR-inefficient

has θ * < 1 (CCR-inefiicient).Then there must be at least one
constraint (or DMU) in (2.9) for which the weight
{v*,u*) produces equality between the left and right hand sides
since, otherwise,θ * could be enlarged
Thus, CCR-inefRciency means that either (i) 9* < 1 or (M) 6* =
1 and at least one element of {v*,u*) is zero for every optimal
solution of (LPo).
2.6.1 Example 2.1 (1 Input and 1 Output
Case)
DMU A
Input
Output

B
2
1
C
3
3
D
3
2
E
4
3
F
5
4
G
5
2
H
6
3
We can evaluate the efficiency of DMU A, by
solving the LP problem below
The optimal solution,
easily obtained by
simple ratio
calculations, is given
by(v* = 0.5, u* = 0.5,
e* = 0.5).
8
5
DEA-Solver Pro5.0- CCR-I
No.
DMU
1A
2B
3C
4D
5E
6F
7G
8H
Score
V(1) Input
U(1)
Output
0.5
1
0.6666667
0.75
0.8
0.4
0.5
0.625
0.5
0.3333333
0.3333333
0.25
0.2
0.2
0.1666667
0.125
0.5
0.3333333
0.3333333
0.25
0.2
0.2
0.1666667
0.125
The θ * values in Table 2.2 show what is needed to bring each
DMU onto the efficient frontier. For example, the value of
6* = 1/2 applied to A's input will bring A onto the efficient
frontier by reducing its input(2) 50% while leaving its output at
its present value(1).
No.
DMU
1A
2B
3C
4D
5E
6F
7G
8H
Score
Rank
0.5
1
0.6666667
0.75
0.8
0.4
0.5
0.625
Reference set (lambda)
6B
0.3333333
1B
1
4B
0.6666667
3B
1
2B
1.3333333
8B
0.6666667
6B
1
5B
1.6666667
Example 2.2
DMU A
B
C
D
E
F
Input Xl
4
7
8
4
2
10
X2
3
3
1
2
4
1
1
1
1
1
1
1
Output y
Example -2.2DEA-Solver Pro5.0/
CCR(CCR-I)
No.
DMU
1A
2B
3C
4D
5E
6F
Score
0.8571429
0.6315789
1
1
1
1
V(1) Xl
V(2) X2
0.14285714 0.1428571
5.26E-02 0.2105263
8.33E-02 0.3333333
0.16666667 0.1666667
0.5
0
0
1
U(1) y
0.8571429
0.6315789
1
1
1
1
The (unique) optimal solution is (v*= 0.0526, v* = 0.2105,
u* = 0.6316, 0* =0.6316), the CCR-efRciency of B is 0.6316,
and the reference set is EB ={C, D}.
Example -2.2DEA-Solver Pro5.0/
CCR(CCR-I)
No.
DMU
1A
2B
3C
4D
5E
6F
Score
0.857142
9
0.631578
9
1
1
1
1
Rank
Reference set
(lambda)
0.714285
5D
E
7
0.105263
6C
D
2
1C
1
1D
1
1E
1
1C
1
0.285714
3
0.894736
8
Now let us observe the difference between the
optimal weights v* = 0.0526and v^ = 0.2105.The
ratio v^/v^ = 0.2105/0.0526 = 4 suggests that it is
advantageous for B to weight Input X2 four times
more than Input xi in order to maximize the ratio
scale measured by virtual input vs. virtual output.
 Therefore a reduction in Input X2 has a bigger
effect on efficiency than does a reduction in Input
X1

The optimal solution for F is (v1*= 0, v2* = 1, u*
=1, 0* =1) and with 0* = 1, F looks efficient.
However, we notice that v* = 0.
 Furthermore, let us examine the inefficiency of F
by comparing F with C. C has Input x1 = 8 and Input
x2 = 1, while F has Input xi = 10 and Input X2 =1. F
has 2 units of excess in Input X1 compared with C.
 This deficiency is concealed because the optimal
solution forces the weight of Input Xi to zero (v^ =
0). C is in the reference set of F and hence by direct
comparison we can identify the fact that F has
used an excessive amount of this input.
