=Paper=
{{Paper
|id=Vol-1623/paperapp8
|storemode=property
|title=Malmquist Productivity Index for Network Production Systems
|pdfUrl=https://ceur-ws.org/Vol-1623/paperapp8.pdf
|volume=Vol-1623
|authors=Chiang Kao
|dblpUrl=https://dblp.org/rec/conf/door/Kao16
}}
==Malmquist Productivity Index for Network Production Systems==
https://ceur-ws.org/Vol-1623/paperapp8.pdf
Malmquist productivity index
for network production systems
Chiang Kao
Department of Industrial and Information Management
National Cheng Kung University, Tainan 701, Taiwan
e-mail: ckao@mail.ncku.edu.tw
Abstract. The conventional Malmquist productivity index (MPI) measures the
performance improvement of a production system between two periods, where
the system is treated as a black box, ignoring the internal operations of the
component processes. Based on a relational model of the data envelopment
analysis (DEA) for measuring the system and process efficiencies, this paper
develops a methodology for calculating the system and process MPIs in one
model. Moreover, relationships between the system and process MPIs are de-
rived. By defining the ratio of the inefficiencies of a unit in two periods as
complementary MPIs, this paper finds that the system complementary MPI is a
linear combination of the process complementary MPIs, and the former is also
close to a weighted average of the latter. Knowing the relationship between the
system and process MPIs helps identify the processes that deter the improve-
ment of the system; amendments to them will improve the system performance
in the future.
Keywords: data envelopment analysis, Malmquist productivity index, network
system.
1 Introduction
Efficiency measurement is important for organizations to identify unsatisfactory
operations so that making improvements to them will produce more outputs with the
same amount of inputs. If an organization is relatively efficient as compared to other
similar ones at a point of time, yet it is actually declining as compared to its past per-
formance, certain indexes for alerting the decision maker are necessary. One such
index is the Malmquist productivity index (MPI), which measures efficiency changes
between two periods for an organization, or any decision making unit (DMU).
The MPI has been widely applied to measuring performance changes between two
periods, especially due to an act or policy (Banker et al., 2005; Chang et al., 2009;
Kao, 2000). Different forms of the MPI have been proposed in the literature. Suppose
the efficiency change of a DMU between periods t and t+1 is to be measured. The
early work of Caves et al. (1982a, 1982b) calculated the relative efficiencies of the
two periods based on the production technology of period t. Since the production
Copyright © by the paper's authors. Copying permitted for private and academic purposes.
In: A. Kononov et al. (eds.): DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org
734 Chiang Kao
technology of period t+1 can also be used for calculating the relative efficiencies, and
the results are probably different from those calculated from the technology of period
t, Färe et al. (1994) suggested using the geometric mean of the two measures as the
MPI to solve the problem of disparity.
A system is usually composed of several processes connected in a network struc-
ture. The conventional MPI measures the performance improvement of a system con-
sidered as a whole unit, neglecting the operations of its component processes. How-
ever, for a network system, it is possible that some processes are worsened while the
system is improved. It is also possible that the system is worsened, while some pro-
cesses are improving. Merely looking at the system MPI cannot identify the processes
that cause the deterioration of the aggregate performance. The objective of this paper
is to develop a methodology for measuring the system and process MPIs at the same
time, and explore the relationship between them so that unsatisfactory processes can
be identified. The MPI used for discussion is the global MPI.
The basic component of MPI is relative efficiency, and the data envelopment anal-
ysis (DEA) technique (Charnes et al., 1978) has been widely used for its calculation.
To calculate the MPI of a network system, a network DEA model is needed. Various
network DEA models have been developed in the literature (see, for example, the
review of Kao (2014)), and they can be classified into three types: independent, con-
nected (or descriptive), and relational. For independent models, the system and indi-
vidual process efficiencies are calculated independently and separately, without con-
sidering the relations among them.
For descriptive models, the operations of all component processes are described in
the model in calculating the system efficiency. The results obtained are more reasona-
ble; nevertheless, the process efficiencies still need to be calculated separately. The
system and process efficiencies do not have any relationship, either. The third type of
models, relational, on the other hand, takes the relations between the system and pro-
cesses into consideration in developing the model. The system and process efficien-
cies can be calculated at the same time. Moreover, there exist some mathematical
relationships between the system and process efficiencies (Chen et al., 2009a; 2009b;
Kao and Hwang, 2008). Due to this property, this paper uses the relational model to
calculate efficiencies. Based on the mathematical relationship between the system and
process efficiencies, certain relationships between the system and process MPIs will
be derived.
In the next section, relational models for different network structures are firstly re-
viewed. Then, in Section 3, models for calculating the global MPI for network sys-
tems are developed. Relationships between the system and process MPIs are investi-
gated. After that, Section 4 uses three examples to explain the methodology proposed
in this paper. Finally, in Section 5, some conclusions are drawn.
2 The Relational Model
Consider a set of n DMUs, each uses the same m inputs to produce the same s out-
puts. Denote Xij and Yrj as the ith input, i=1,…, m, and rth output, r=1,…, s, respec-
Malmquist productivity index for network production systems 735
tively, of the jth DMU, j=1,…, n. The CCR model of DEA for calculating the effi-
ciency of DMU k under the assumption of constant returns-to-scale can be formulated
as (Charnes et al., 1978):
s
Ek = max. urYrk
r 1
m
s.t. vi X ik 1
i 1
s m
urYrj vi X ij 0, j 1, ...,n (1)
r 1 i 1
ur , vi ε, r 1, ..., s, i 1, ...,m
where ur and vi are virtual multipliers and is a small non-Archimedean number
(Charnes and Cooper, 1984) imposed to prevent any input/output factor from being
ignored in calculating the efficiency.
Usually a system is composed of several processes connected as a network. Model
(1) treats the system as a black-box, neglecting the operations of the component pro-
cesses. Consequently, it is possible that all processes are not efficient while the sys-
tem, as a whole, is. The network DEA takes the operations of the processes into con-
sideration in calculating the system efficiency so that unreasonable results can be
excluded. Network systems have various structures. The two fundamental ones are
series and parallel.
The series structure is a basic network structure where a number of processes are
connected in series. The characteristic of this type of structure is that the inputs used
by all processes, except the first, are produced by their preceding one, and the outputs
produced by all processes, except the last, are utilized by their succeeding one. The
series structure is the most widely discussed network structure in the DEA literature.
Let Z (fjp ) denote the fth intermediate product, f=1,…, g, produced by process p,
p=1,…, q1. Note that the intermediate products produced by the last process, q, are
the outputs of the system, Yrj. Kao and Hwang (2008) showed that the system effi-
ciency is the product of the q process efficiencies:
q s
Ek( p ) ur*Yrk EkS (2)
p 1 r 1
The parallel structure is another basic network structure which is composed of a
number of q processes, and each applies inputs X ij( p ) to produce outputs Yrj( p ) . The
total inputs consumed by the system are Xij p 1 X ij( p ) , and the total outputs pro-
q
duced are Yrj p 1Yrj .
q ( p)
The relational model assigns the same multiplier to the same factor, regardless of
which process it is associated with. Kao (2009) showed that the system efficiency is a
weighted average of the q process efficiencies:
u r Yrk q s
s * ( p)
q q
m
( p ) Ek( p ) vi* X ik( p ) mr 1 * ( p ) ur*Yrk( p ) EkS (3)
p 1 p 1 i 1 i 1 vi X ik p 1 r 1
where ( p ) = i 1 vi* X ik( p ) / i 1 vi* X ik = i 1 vi* X ik( p ) .
m m m
736 Chiang Kao
Most network systems are a mixture of the series and parallel structures. Theoreti-
cally, they can have numerous forms of structure, although the most complicated
structure that appears in the literature only has five processes (Lewis and Sexton,
2004). Denote I (j p ) , O (j p ) {1, 2,…, g} as the index sets of the input and output in-
termediate products, respectively, of process p for DMU j. To be generic, we consider
the very general network structure shown in Figure 1, where each process p consumes
exogenous inputs X ij( p ) and intermediate products Z (fjp ) , j I (j p ) that are produced by
other processes to produce exogenous outputs Yrj( p ) and intermediate products Z (fjp ) ,
j O (j p ) for other processes to use. The total inputs consumed and the total outputs
produced by the system are Xij p 1 X ij( p ) and Yrj p 1Yrj( p ) , respectively. Let the
q q
same factor have the same multiplier; the relational model for calculating the system
efficiency is:
s
EkS max. urYrk
r 1
m
s.t. vi X ik 1
i 1
s m
ur Yrj vi X ij s j 0, j 1, ...,n (4)
r 1 i 1
s m
( u r Yrj( p ) w f Z (fjp ) ) ( vi X ij( p ) w f Z (fjp ) ) s (j p ) 0,
r 1 f O ( p)
j i 1 f I (j p )
j 1, ...,n, p 1, ...,q
ur , vi , w f ε, r 1, ..., s, i 1, ...,m, f 1, ..., g
sj, s ( p)
j 0, j 1, ...,n, p 1, ...,q
Since all the intermediate products are produced and consumed in the system, the
sum of the constraints corresponding to the q processes is equal to the constraint cor-
responding to the system for each DMU. X i( p ) i=1,…, m
Z (f p ) Z (f p )
p
f I ( p) f O( p)
…
…
Yr( p )
r=1,…s
, s
Fig. 1. The general network structure.
Based on Model (4), the system and process efficiencies of DMU k can be calcu-
lated as:
s m s
EkS u r*Yrk / vi* X ik u r*Yrk (5)
r 1 i 1 r 1
Malmquist productivity index for network production systems 737
s m
Ek( p ) [ u r*Yrk( p ) w*f Z (fkp ) ] /[ vi* X ik( p ) w*f Z (fkp ) ], p 1, ...,q
r 1 f Ok( p ) i 1 f I k( p )
where ( u r* , vi* , w*f ) is a set of optimal solutions. Since the sum of the q process
constraints is equal to the system constraint for each DMU, that is, s sk p 1 sk( p ) ,
q
we have: r 1 ur Yrk i 1 vi X ik p 1[r 1 urYrk
s * m * q s * ( p)
f O w f Z fk ] )
* ( p)
( p)
i 1 i ik
k
m * ( p) * ( p) S ( p)
v X f I w f Z
( p)
k
fk )] , or, from the expression of E k and Ek in Equation
(9):
q m q
1 EkS (1 Ek( p ) )( vi* X ik( p ) w*f Z (fkp ) ) (1 Ek( p ) ) ( p ) (6)
p 1 i 1 f I k( p ) p 1
where ( p ) i 1 vi* X ik( p ) f I w*f Z (fkp ) . That is, the system inefficiency, 1 EkS ,
m
( p)
k
is a linear combination of the q process inefficiencies, 1 Ek( p ) . Nevertheless, the
former is not a weighted average of the latter because the sum of the weights,
p 1 ( p ) p 1 [i 1 vi X ik f I w*f Z fk ] 1 p 1 f I w*f Z (fkp ) , is
q q * ( p) m ( p) q
( p) ( p)
k k
clearly greater than 1.
3 Malmquist Productivity Index
The MPI is an index for representing the efficiency improvement between two pe-
riods. Various forms of MPI have been developed. The global MPI proposed by Pas-
tor and Lovell (2005) has several attractive properties and is used in this paper. The
basic idea of the global MPI is to use the observations of all periods to construct the
production frontier. Based on which, the relative efficiencies of a DMU in two peri-
ods are calculated, and their ratio is the MPI. Since all observations have been includ-
ed in constructing the frontier, the resulting efficiencies for both periods will not ex-
ceed one.
Let the superscript h, t, and t+1, denote period. The relational model for calculating
the relative efficiency of DMU k in period t+1 for the series structure based on the
technology constructed from the observations of periods t and t+1 is:
s
( EkS )t 1 max. ur (Yrk ) t 1
r 1
m
s.t. vi ( X ik )t 1 1
i 1
g m
w f (Z (fj1) )h vi ( X ij )h 0, h t, ...,t 1, j 1, ...,n (7)
f 1 i 1
g g
w f Z ((fjp ) )h w f (Z (fjp 1) )h 0, h t, ...,t 1,
f 1 f 1
j 1, ...,n , p 2, ...,q 1
s g
ur (Yrj )h w f (Z (fjq 1) )h 0, h t, ...,t 1, j 1, ...,n
r 1 f 1
738 Chiang Kao
ur , vi , w f ε, r 1, ..., s, i 1, ...,m, f 1, ..., g
The system MPI for DMU k, M kS , is the ratio of ( EkS )t 1 to ( EkS ) t . Kao and
Hwang (2014) showed that it can be expressed as:
( EkS )t 1 p 1 ( Ek )
q ( p ) t 1 q
( Ek( p ) )t 1 q
M kS ( p ) t M k( p ) (8)
p 1 ( Ek )
S t q ( p) t
( Ek ) p 1 ( Ek ) p 1
where M k( p ) ( Ek( p ) )t 1 / ( Ek( p ) ) t is, by definition, the MPI of process p. Hence, the
system MPI is the product of the process MPIs.
The way of calculating the system MPI for the parallel structure is similar to that
for the series structure. The model for calculating the system efficiency of period t+1
based on the combined technology is:
s
( EkS )t 1 max. ur (Yrk ) t 1
r 1
m
s.t. vi ( X ik )t 1 1
i 1
s m
ur (Yrj( p ) )h vi ( X ij( p ) )h 0, h t, ...,t 1,
r 1 i 1
j 1, ...,n , p 1, ...,q (9)
ur , vi ε, r 1, ..., s, i 1, ...,m
According to Equation (3), the system efficiency for the parallel structure is a
weighted average of the process efficiencies: ( EkS )h p 1[i 1 vih ( X ik( p ) ) h ]( Ek( p ) ) h ,
q m
h t, ...,t 1, where ( EkS )h r 1 urh (Yrk )h and ( Ek( p ) )h r 1 urh (Yrk( p ) )h / im1 vih ( X ik( p ) )h .
s s
The system MPI can then be expressed as (Kao 2016):
( EkS )t 1 p 1[i 1 vi ( X ik ) ]( Ek )
q m t 1 ( p ) t 1 ( p ) t 1
M kS
( EkS )t ( EkS )t
( p ) t 1
( E ) ( Ek )t
q m ( p)
vit 1 ( X ik( p ) )t 1 k ( p ) t
p 1 i 1 ( Ek ) ( EkS )t
u t (Y ( p ) )t / vt ( X ( p ) )t
s m
q
m
vit 1 ( X ik( p ) )t 1 M k( p ) r 1 r rk s t i 1 t i ik
p 1 i 1 ( pr)1 ur (Yrk )
q
s
u t
(Y ( p) t
)
m
v t 1
( X ik )
t 1 q
M k( p ) rs1 t rk tr i 1 i
M k( p ) ( p ) (10)
u (Y ) m vt ( X ( p ) )t p 1
p 1
r 1 r rk i 1 i ik
A result that the system MPI is a linear combination of the process MPIs is de-
rived. However, the former is not a weighted average of the latter because the sum of
the weights, p 1 ( p ) , is not necessarily equal to 1.
q
Theoretically, if the total weight, p 1 ( p ) , in a linear combination is much
q
smaller than 1, then it is possible that all process MPIs are greater than 1, yet their
linear combination is less than 1. The opposite situation may also happen, if the total
weight is much greater than 1. In practice, the chance of having a very large or a very
small total weight is quite low. The reason is because the observations of DMU k for
each process p in period t, ( ( X ik( p ) )t , (Yrk( p ) ) t ), and period t+1, ( ( X ik( p ) )t 1 , (Yrk( p ) )t 1 ),
are usually not too different, which make i 1 vit 1 ( X ik( p ) )t 1 approximately equal to
m
Malmquist productivity index for network production systems 739
i 1 vit ( X ik( p ) )t , especially when they are under the same frontier facet where vit = vit 1 .
m
In this case, the ratio of i 1 vi ( X ik( p ) )t 1 to i 1 vit ( X ik( p ) )t in ( p ) is approximately
m t 1 m
equal to 1, which simplifies ( p ) to ˆ ( p ) = r 1 urt (Yrk( p ) )t / r 1 urt (Yrk )t . Since
s s
p 1ˆ ( p ) 1, a result in which the system MPI is approximately equal to a weighted
q
average of the process MPIs is obtained, where the weight associated with process p
is the proportion of the aggregate output of this process in all processes.
For network systems with a general structure of Figure 1, Model (4) can be
adapted to suit the combined technology for calculating the efficiency of DMU k in
period t+1:
s
( EkS )t 1 max. ur (Yrk ) t 1
r 1
m
s.t. vi ( X ik )t 1 1
i 1
s m
ur (Yrj )h vi ( X ij )h (s j )h 0, h t, ...,t 1, j 1, ...,n (11)
r 1 i 1
s m
[ ur (Y )
( p) h
rj w f (Z (fjp ) )h ] [ vi ( X ij( p ) )h w f (Z (fjp ) )h ]
r 1 f O ( p)
j i 1 f I (j p )
(s j ) 0, h t , ...,t 1, j 1, ...,n , p 1, ...,q
( p) h
ur , vi , w f ε, r 1, ..., s, i 1, ...,m, f 1, ..., g
( s j ) h , (s (j p ) )h 0, h t, ...,t 1, j 1, ...,n , p 1, ...,q
From Equation (6), the system inefficiency is a linear combination of the process
inefficiencies: 1 ( EkS )h p 1 (1 ( Ek( p ) ) h )( ( p ) ) h for h t, ...,t 1, where ( ( p ) )h
q
i 1 vi ( X ik ) f I w f (Z (fkp) )h . The conventional MPI is defined as the ratio of
m h ( p) h h
( p)
k
the efficiencies of two periods. A value greater (or less) than 1 indicates that the effi-
ciency has improved (or is worsened). Conceptually, the MPI can also be defined as
the ratio of the inefficiencies of two periods. Under this definition, a value less (or
greater) than 1 indicates that the efficiency has improved(or is worsened). Term this
type of MPI the complementary MPI, and denote it as M . The system complemen-
tary MPI for DMU k can be expressed as:
1 ( EkS )t 1 p 1[1 ( Ek ) ][ i 1 vi ( X ik ) f I w f ( Z fk ) ]
q( p ) t 1 t 1 ( p ) t 1m t 1 ( p ) t 1
( p)
M kS k
1 ( EkS )t 1 ( EkS )t
q
1 ( Ek )t 1 [1 ( Ek ) ][ i 1 vit 1 ( X ik( p ) ) t 1 f I wtf1 ( Z (fkp ) ) t 1 ]
( p ) ( p ) t m
( p)
( p) t
k
p 1 1 ( Ek ) 1 ( EkS ) t
From the constraints of Model (11), we have 1 ( Ek ) (sk ) (because S t t
i 1 vi ( X ik )t =1) and 1 ( Ek( p ) )t (sk( p ) )t / i 1 vit ( X ik( p ) )t f I wtf (Z (fkp) )t ]. Denote
m m
( p)
k
(s ( p) t
) [ v ( X
m t 1
i 1 i
( p ) t 1
) f I w ( Z t 1 ( p ) t 1
) ]
( p)
k ik ( p)
k
f fk
( s ) [ v ( X
k
t m t
i 1 i ik ) f I w ( Z
( p) t
( p)
t
f
( p) t
fk )]
k
M kS becomes:
q
M kS M k( p ) ( p ) (12)
p 1
740 Chiang Kao
where M k( p ) [1 ( Ek( p ) )t 1 ] /[1 ( Ek( p ) )t ] is the complementary MPI of process p.
Here a property, that the system complementary MPI is a linear combination of the
process complementary MPIs, is obtained.
For cases where ( EkS )t 1, M kS is undefined. However, since ( EkS )t 1 implies
( Ek( p ) )t 1 for all p, the system and process MPIs become: M kS ( EkS )t 1 and
M k( p ) ( Ek( p ) )t 1 . The relationship between the system and process MPIs then is:
q m
1 M kS 1 ( EkS )t 1 (1 M k( p ) ) vit 1 ( X ik( p ) )t 1 wtf1 ( Z (fkp ) ) t 1
q
p 1 i 1 f I( p)
k
(1 M k ) ( p) ( p)
(13)
p 1
The value of M kS in this case is always less than or equal to 1, indicating that the
performance is worsened. If, for period t, the system is not efficient, yet a process is,
then the corresponding term M k( p ) in Equation (12) is changed to [1 M k( p ) ]
[im1 vit 1 ( X ik( p ) )t 1 f I wtf1 (Z (fkp ) )t 1 ] / ( sk ) t .
( p)
k
4 Example
In this section we use an example to explain how the models developed in Section
3 are used to calculate the system and process MPIs, and the relationships between
them.
Consider a system of three processes connected in a general structure shown in
Figure 2, where process 1 applies input X1 to produce output Y1, process 2 applies
input X2 to produce output Y2, and process 3 applies input X3 and portions of Y1 and Y2
to produce output Y3. There are eight DMUs to be compared, with the data shown in
Table 1. Note that the output of process 1, Y1, is split into Y1( I ) and Y1(O ) , where the
former is used by process 3 for producing Y3 and the latter is an output of the system.
Similarly, Y2, is split into Y2( I ) and Y2(O ) .
X1 1 Y1(O )
Y1( I )
X3 3 Y3
(I )
Y 2
X2 2 Y2(O )
Fig. 2. Structure of the general network example.
By applying Model (11), the system and process inefficiencies are calculated as
shown in the first two rows of each DMU in Table 2. As discussed in Section 2, the
system inefficiency is a linear combination of the process inefficiencies, and the sum
Malmquist productivity index for network production systems 741
of the weights for linear combination is greater than 1 for all DMUs (shown in the last
column in parentheses). The ratio of the inefficiencies in periods t+1 and t is the com-
plementary MPI, and is shown in the third row and denoted as cMPI. The weight
associated with each process is calculated from Equation (12) shown in parentheses
next to the complementary MPI. The system complementary MPI is a linear combina-
tion of the process complementary MPIs.
Table 1. Data for the general structure example.
DMU Period X1 X2 X3 Y1 ( Y ( I ) Y (O ) ) Y2 ( Y ( I ) Y (O ) ) Y3
1 1 2 2
1 t 2 4 3 5 (2 3) 5 (2 3) 4
t+1 2 4 2 6 (3 3) 5 (2 3) 4
2 t 3 5 3 5 (3 2) 6 (3 3) 5
t+1 2 5 4 5 (3 2) 7 (3 4) 6
3 t 3 6 3 6 (2 4) 6 (2 4) 5
t+1 4 7 4 7 (3 4) 8 (4 4) 7
4 t 4 6 4 7 (3 4) 6 (3 3) 7
t+1 3 5 3 7 (3 4) 6 (3 3) 6
5 t 5 6 4 7 (4 3) 7 (3 4) 7
t+1 4 5 4 7 (4 3) 6 (3 3) 7
6 t 5 7 5 8 (4 4) 8 (3 5) 8
t+1 5 7 6 9 (4 5) 9 (4 5) 9
7 t 5 8 5 9 (4 5) 9 (4 5) 9
t+1 6 7 5 9 (4 5) 9 (5 4) 10
8 t 6 9 5 9 (5 4) 9 (4 5) 8
t+1 5 8 4 9 (5 4) 9 (5 4) 9
A complementary MPI of greater (or less) than one indicates that the performance
of the associated unit is worsened (or has improved). When the efficiency of period t,
( Ek( p ) ) t , is close to 1, the inefficiency, 1 ( Ek( p ) )t , will be close to 0, which may re-
sult in peculiar numbers in calculating ratios. The complementary MPIs of process 3
of DMUs 3 and 6 are examples where the former has a value of 391.75 and the latter
has a value of 674.76. These large values are difficult for human to interpret how
worse the units have performed. A contrary situation is when ( Ek( p ) )t 1 is close to 1,
which makes the complementary MPI close to 0. For example, both process 1 of
DMU 1 and process 2 of DMU 2 have a complementary MPI of 0. There is no way of
knowing which one has improved more.
Since the MPI and the complementary MPI have opposite trends intersecting at 1,
one can rely on the MPI to help judge the extent of the performance change. The
fourth row of Table 2 shows the MPI, which is the ratio of E(t+1)/E(t), of each DMU.
Comparing the complementary MPI in the third row with the MPI in the fourth, it is
clear that when one has a value greater (or less) than 1, the other has a value less (or
greater) than 1. For the example of 391.75 and 674.76 mentioned in the preceding
paragraph, we know that both processes are worsened, yet their extents are difficult to
judge. Their corresponding MPIs of 0.8753 and 0.90021 make the judgment much
easier. Similarly, for process 1 of DMU 1 and process 2 of DMU 2, where both pro-
cesses have a complementary MPI of 0, the former has an MPI of 1.2 and the latter an
MPI of 1.1667, indicating that the former has improved more than the latter. Hence,
one may use the MPI to judge the extent of performance changes. The complementary
MPI is merely for showing the mathematical relationship between the performance
change of the system and the processes.
742 Chiang Kao
Table 2. Inefficiencies, complementary MPIs, and MPIs of the general structure example.
DMU Process 1 Process 2 Process 3 System
Score (weight) Score(weight) Score(weight) Score(weight)
1 1E(t) 0.1667 (0.0006) 0.1071 (0.6516) 0.2000 (0.5807) 0.1861 (1.2329)
1E(t+1) 0.0000 (0.0006) 0.1071 (0.7360) 0.0003 (0.5266) 0.0790 (1.2632)
cMPI 0.0000 (0.0005) 1.0000 (0.4238) 0.0013 (0.5660) 0.4246 (0.9904)
MPI 1.2000 1.0000 1.2497 1.1315
2 1E(t) 0.4444 (0.0009) 0.1429 (0.6988) 0.1668 (0.6001) 0.2003 (1.2998)
1E(t+1) 0.1667 (0.0006) 0.0000 (0.6370) 0.1430 (0.6357) 0.0910 (1.2733)
cMPI 0.3750 (0.0013) 0.0000 (0.4544) 0.8575 (0.5292) 0.4543 (0.9849)
MPI 1.5000 1.1667 1.0285 1.1367
3 1E(t) 0.3333 (0.0009) 0.2857 (0.7356) 0.0003 (0.4389) 0.2106 (1.1753)
1E(t+1) 0.4167 (0.0012) 0.1837 (0.7085) 0.1250 (0.5798) 0.2031 (1.2895)
cMPI 1.2500 (0.0019) 0.6429 (0.9612) 391.75 (0.0009) 0.9643 (0.9640)
MPI 0.8750 1.1429 0.8753 1.0095
4 1E(t) 0.4167 (0.0012) 0.2857 (0.6758) 0.0003 (0.5646) 0.1938 (1.2417)
1E(t+1) 0.2222 (0.0009) 0.1429 (0.6988) 0.0001 (0.6001) 0.1001 (1.2998)
cMPI 0.5333 (0.0019) 0.5000 (1.0304) 0.3521 (0.0009) 0.5165 (1.0332)
MPI 1.3333 1.2000 1.0002 1.1162
5 1E(t) 0.5333 (0.0015) 0.1667 (0.6764) 0.0002 (0.5641) 0.1137 (1.2420)
1E(t+1) 0.4167 (0.0012) 0.1429 (0.6356) 0.0002 (0.6360) 0.1914 (0.9932)
cMPI 0.7813 (0.0056) 0.8571 (0.9321) 0.8867 (0.0012) 0.8044 (0.9389)
MPI 1.2500 1.0286 1.0000 1.0251
6 1E(t) 0.4667 (0.0015) 0.1837 (0.6612) 0.0001 (0.5401) 0.1222 (1.2028)
1E(t+1) 0.0400 (0.0015) 0.0816 (0.6193) 0.1000 (0.6324) 0.1144 (1.2532)
cMPI 0.8571 (0.0057) 0.4444 (0.9307) 674.76 (0.0008) 0.9360 (0.9372)
MPI 1.1250 1.1250 0.9001 1.0089
7 1E(t) 0.4000 (0.0015) 0.1964 (0.6894) 0.0003 (0.5557) 0.1362 (1.2466)
1E(t+1) 0.5000 (0.0018) 0.0816 (0.6600) 0.0000 (0.6753) 0.0548 (1.3372)
cMPI 1.2500 (0.0053) 0.4156 (0.9520) 0.0000 (0.0016) 0.4022 (0.9733)
MPI 0.8333 1.1429 1.0003 1.0942
8 1E(t) 0.5000 (0.0018) 0.2857 (0.7136) 0.1116 (0.5116) 0.2619 (1.2432)
1E(t+1) 0.4000 (0.0015) 0.1964 (0.0011) 0.0000 (0.9984) 0.0008 (1.0010)
cMPI 0.8000 (0.0029) 0.6875 (0.0012) 0.0000 (0.4254) 0.0031 (0.4295)
MPI 1.2000 1.1245 1.1256 1.3537
5 Conclusion
A system is usually composed of several processes operating interdependently. The
conventional MPI measures the efficiency improvement between two periods by
treating the system as a black box, neglecting the operations of the component pro-
cesses. Once the MPI is calculated, it is not clear which processes cause the im-
provement of deterioration of the system. More seriously, it is possible that the system
shows an improvement while most of the component processes are actually worsened.
This paper adopts the idea of the relational DEA model to calculate the system and
process efficiencies at the same time. Most importantly, there exist mathematical
relationships between them. Based on the relationships between the system and pro-
cess efficiencies, relationships between those of MPIs are derived. The relationships
show the effects of the process MPIs on the system MPI, and the processes which
Malmquist productivity index for network production systems 743
cause the improvement or deterioration of the system performance can also be identi-
fied.
By defining the complementary MPI as the ratio of the inefficiencies of two peri-
ods, this paper finds that the system complementary MPI is a linear combination of
the process complementary MPIs, and the former is approximately a weighted aver-
age of the latter. Although the complementary MPI can also be used to judge whether
a unit has improved or not between two periods, its magnitude is difficult to interpret
as was illustrated by the general-structure example. For these cases, the conventional
MPI can be calculated to help the interpretation.
The DEA model used in this study is the CCR model under the assumption of con-
stant returns-to-scale. For series systems with only two processes, Kao and Hwang
(2011) were able to use the BCC model (Banker et al., 1984) to calculate the efficien-
cy under the assumption of variable returns-to-scale. Its extension to the calculation
and decomposition of MPI should be straightforward. For parallel systems, replacing
the CCR model by the BCC model is not difficult, as briefly mentioned in Kao
(2009). How to generalize to cases of variable returns-to-scale for series structures
with more than two processes and general network structures are not so simple, and is
a direction for future research.
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