Asia Pacific Management Review (2006) 11(6), 389-392
Explore the Optimal Allocation Problem with AHP/DEA
Tzai-Zang Lee * and Ya-Fen Tseng
Department of Industrial and Information Management National Cheng Kung University, Tainan, Taiwan
Accepted in November 2006
Available online
Abstract
This paper presents the optimal allocation for the human resource practices in DMUs (Decision-making units) where each unit has
multiple inputs and outputs, the units is classified into two groups: efficient and inefficient. From the literature reviews, scientists suggest a
“best allocation approach” without seriously taking into account differences. The question arises the issue which this study would like to
explore. This paper deems that the human resource practices are the inputs and the measures of organizational performance are the outputs.
From the AHP/DEA (Analytical Hierarchical Process/ Data Envelopment Analysis), it can be found which units have the efficiency. But
few papers explore which the variable combination of the human resource and organizational performance can make the efficiency. This
study continues to look for the weight of the variables. By the weight of the input and output variables, this study presents the allocation of
the human resource practices.
Keywords: Analytical hierarchical process; Data envelopment analysis; Optimal allocation
1. Introduction
2. Literature Review
This paper presents the optimal allocation of human
resource practices on organization performance in DMUs
(Decision-making units) where each unit has multiple inputs and outputs, the units are classified into two groups:
efficient and inefficient. When the variables of the input
and output in each DMU (Decision-making unit) are selected, each of the writers orders the ranking according to
five-point Likert scales. And, that 1 to 5 meant “strongly
disagree” to “strongly agree.”
Data Envelopment Analysis has become a powerful
tool for evaluating efficiency of decisions-making units
(Ruggiero, 2004). The linear programming model is introduced and extended by Charnes et al (1978) and Banker et
al (1984). Since DEA compares with DMUs observed outputs and inputs, DEA is applied cross-evocatively and the
efficiency measures are averaged (Ruggiero, 2004). Then,
analytic hierarchy process is also becoming quite popular
in research. The development of AHP could be traced
back to the early 1970s (Satty, 1980). As the method procedure of AHP is easily combined with multiple, objective
programming formulations with interactive solution process (Yang and Lee, 1997), it has obtained a wider attention
from some literature. AHP considers both qualitative and
quantitative approaches to research and combines them
into a single experimental exploration.
Although DEA is originally designed for classification,
it is very often required for measuring overall relative productivity, efficiency and comparing units in many allocations. It is extremely important to compare the uniformity
between various evaluation methods and to take the best of
each (Friedman and Sinuany-Stern, 1998). The advantage
of the AHP/DEA ranking model is that the AHP (Analytical Hierarchical Process) pairwise comparisons are derived mathematically from the input/output data by running pairwise DEA runs ( Sinuany-Stern, Mehrez and Hadad, 2000). From the literature review, the authors research the AHP/DEA from the efficiency of DMUs.
Sinuany-Stern et al. (1994) employ the DEA classification to analyze the linear programming analysis for
ranking DMUs. Friedman and Sinuany-Stern (1997) show
the canonical correlation analysis (CCA/DEA) to rank the
full DMUs. Sinuany-Stern and Friedman (1998a, 1998b)
use the pariwise efficiency matrix to sort DMUs. These
authors explore the efficiency based on DEA methodology.
Sinuany-Stern, Mehrez and Hadad (2000) prove that AHP
and DEA have the same characteristic on the single input
and single output. They apply cross evolution concept
(AHP method) into ranking DUMs (DEA method) and
advance the multiple inputs and outputs based on DEA.
Few papers explore the pairwise comparisons and the
importance of input and output variables in all DMUs. In
this paper, it shows optimal allocation of human resource
practices utilizing AHP/DEA method as the case study.
This study clearly finds the optimal allocation of the human resource practices on organizational performance
according the concept of statistical weight displayed the
importance.
*
Email: [email protected]
389
Tzai-Zang Lee and Ya-Fen Tseng/Asia Pacific Management Review (2006) 11(6), 389-392
For the resolving the problem of the relative significance, this study adopts the weight concept to solve the
optimal allocation. Therefore, this paper assumes that the
sum of the weight is one. It would like to solve the optimal
weight of the input and output variables in the units. So, it
has two conditional equations. As followed:
s
E B A = M a x ∑ u r Y rB
r =1
m
∑
s .t.
vi X
i =1
s
∑
(7 )
(8 )
s
s
m
∑
u r Y rA − E A A ∑ v i X
r =1
s
=1
u r Y rB ≤ 1
r =1
∑
iB
i =1
iA
= 0
ur = 1
r =1
∑u = 1
∑v = 1
r
(9 )
r =1
m
(1 0 )
i
i =1
m
∑
vi = 1
i =1
The input and output variables are considered as two
parts. So, the sum of the weight of the inputs is one. The
sum of the weight of the outputs is one. By the conditions,
this study built the Problem BA anew. By this method, it
computes out the weight ( u r , vi ) of the inputs and output
(1 1)
1 ≥ ur ≥ ε ,
1 ≥ vi ≥ ε
In the context, the most authors explore the efficiency
on the research subjects. Few of them focus on the optimal
allocation. Few authors seek the weight of the input variables which influence the output variables when the
DMUs obtain efficiency. For this purpose, this paper presents the measurement for the optimal allocation of human
resource practice on organization performance.
variables. And, this concept is solved
As followed Table 1, it adopts the DMUs as an example and illustrates the weights of the inputs and outputs
by the pairwise comparison.
Table 1. The DEA and the Optimal Allocation Input and
Output Variables by AHP/DEA
3. Method-AHP/DEA Ranking
DMU X 1 j
It shows the relative comparison of input and output
variables in the same unit from the first stage- DEA ranking. This subject of this paper would like to display the
optimal allocation from these input and output variables.
So, this study precedes the cross evaluation among each
other unit. It displays the more integrated significance in
the variables.
obtaining the maximize efficiency. However, in order to
comparatively evaluate unit B using the optimal allocation
of unit A, Zilla, Abraham and Yossi (2000) calculate
s
E BA =
∑
m
u r YrB /
r =1
∑
when
v i X iB
ur ≥ ε , vi ≥ ε .
Zilla,
i =1
Abraham and Yossi (2000) have more than one optimal
solution for the optimal allocation; thus, given the optimal
for unit A, E AA, that can solve the following problem
L
X mj
Y1 j
Y1 j
L
Ysj
v 2 AA M
v 2 BB M
vmAA u1 AA u 2 AA
v mBB u1BB u 2 BB
M
u sAA
A
v1 AA
v1BB
M
u sBB
M
M
M
M
M
M
M
A
v1 NNA
M
M
M
B
v 2 AB M
v 2 BB M
M
u sNNA
u sAB
B
v1 AB
v1BB
vmNN M
M
vmAB u1 AB u 2 AB
vmBB u1BB u 2 BB
M
u sBB
M
M
M
M
M
M
M
M
B
M
M
M
M
M
M
M
C
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
N
v1 NNN
M
M
M
M
M
u sNNN
A
When there are only two DMUs denoted by the input
and output of unit A in this certain organization, the following program problem E AA, makes the DEA run for
X2 j
M
vmNN
M
In the next step, utilizing the statistic concept solves
the optimal allocation of the weights of all inputs X ij and
outputs Yij . As followed Table 2.
according to secure the best pairwise evaluation for unit B
(Oral and Kettani, 1990):
Table 2. The Optimal Allocation of Input and Output
Variables
Problem BA:
s
E B A = M a x ∑ u r Y rB
r =1
m
s .t .
∑
i =1
s
∑
r =1
s
∑
r =1
viX
iB
=1
(4 )
u r Y rB ≤ 1
u r Y rA − E A A
m
i =1
vi X
iA
= 0
X1 j
…
X mj
Y1 j
…
Ynj
Optimal
Allocation
v1
…
vm
vm
…
un
From the cross-evolution, this study can procure the
more objective weight of the inputs and outputs. This
study provides the thinking direction which can look after
(5 )
∑
Variables
(6 )
u r ≥ ε , vi ≥ ε
390
Tzai-Zang Lee and Ya-Fen Tseng/Asia Pacific Management Review (2006) 11(6), 389-392
both sides-the weight and cross-comparison.
allocated for training (hr13), training currently provided is
leading to satisfactory result (hr14), training plans are developed and monitored for all employees (hr15), training
programs are consistently evaluated (hr16), emphasis on
ling-term development (hr17), organization ensures that all
managers consistently document terminations in a legal
manner (hr18), organization has employees participate in a
formal orientation program (hr19), new companies have a
better chance of surviving if all employees receive incentives based on organization-wide performance (hr20), talking about salary issues during performance appraisals
tends to hurt morals and future performance (hr21), most
employees prefer variable pay systems to fixed pay systems (hr22) and surveys that directly ask employees how
important pay is them are likely to overestimate pay’s true
importance in actual decisions (hr23). This study made the
organization performance as the output variables. And it
included that turnover (op1), productivity (op2), corporate
financial performance (op3), perceived organizational performance (op4), perceived market performance (op5), employee performance (op6), innovation (op7) and employee
relations (op8). This paper adopted the thirty-sixteen subjects and advanced the optimal allocation of human resource practice on organization performance.
v1 = ( v1 AA + v1BB + L + v1NNA + L + v1NNN ) N 2
v2 = ( v2 AA + v2 BB + L + v2 NNA + L + v 2 NNN ) N 2
M
u1 = ( u1 AA + u1BB + L + u1NNA + L + u1NNN ) N 2
u2 = ( u2 AA + u2 BB + L + u2 NNA + L + u2 NNN ) N 2
M
Actually, E BA is the optimal pairwise comparison of
unit B. Symmetrically, Problem BB and AB are solved,
and E BB and E BA are calculated. Finally, based on
these results, Zilla, Abraham and Yossi (2000) constructs
the pairwise comparison matrix needed for AHP from the
result of the paired DEA described above, so that for every
pair of units j and k :
a jk =
E jj + E jk
E kk + E kj
and a jj = 1
The matrix A is n × n . This matrix has not been
evaluated subjectively by a decision maker; rather, it is an
objective evaluation, and is calculated from the DEA
pairwise units, which provides cross evaluation, thus allowing each unit to receive its most favorable evaluation
relative to any other unit.
According to the pairwise comparisons in DMUs, the
relative significance is more subjective. And, it displays
the optimal allocation of human resource and organization
performance (Table 3, Table 4). The 0.042480% (hr1),
0.37817% (hr2), 0.73169% (hr3), 42.4706% (hr4),
0.54520% (hr5), 0.13593% (hr6), 1.00606% (hr7),
0.21871% (hr8), 2.72604% (hr9), 0.72321% (hr10),
1.30890% (hr11), 1.12361% (hr12), 0.35089% (hr13),
2.57481% (hr14), 2.45018% (hr15), 0.71251% (hr16),
2.73978% (hr17), 1.44476% (hr18), 3.15694% (hr19),
1.51700(hr20), 4.21550% (hr21), 4.45061% (hr22) and
67.02233% (hr23) of human resource practice influenced
the 0.48082% (op1), 0.74243% (op2), 2.35914% (op3),
0.46132% (op4), 0.61776% (op5), 0.72210% (op6),
2.70158% (op7) and 91.91485% (op8) of organization
performance. The result is the optimal allocation of human
resource practice on organization performance.
This study explores the new concept for ranking scaling units with multiple inputs and multiple outputs in the
DEA and AHP. The advantage of the AHP/DEA ranking
model is that the AHP pariwise comparisons are obtained
from the input and output data by running pairwise DEA.
And, this paper is no subjective estimate. Moreover, this
research can obtain the more objective weight of the human resource practice and organization performance by
the cross evaluation from the AHP/DEA.
4. The Case of Optimal Allocation
According to the literature review, it shows that the
human resource management has the significant influence
on the organization performance. This study makes the
human resource management as the input variables It includes that regular use of performance appraisal (hr1), part
of pay related to individual performance (hr2), the performance review process is linked to compensation plans
(hr3), the performance review process is standardized and
documented (hr4), promotion and pay increases are based
on achieving documented (hr5), keeps employee wellperform (hr6), actively tries to make jobs as interesting
and varied as possible (hr7), actively uses team working
where possible (hr8), conducted a company-wide attitude
survey in the part two years (hr9), actively implements
equal opportunities practices (hr10), has range of familyfriendly practices in place (hr11), has a works council or
consultative process in place (hr12), sufficient money is
Table 3. The Optimal Allocation of Human Resource
Practices Weight
hr
%
hr
%
hr
%
hr
%
hr
%
391
hr1
0.04248
hr6
0.13593
hr11
1.30890
hr16
0.71251
hr21
4.21550
hr2
0.37817
hr7
1.00606
hr12
1.12361
hr17
2.73978
hr22
4.45061
hr3
0.73169
hr8
0.21871
hr13
0.35090
hr18
1.44476
hr23
67.02233
hr4
0.42471
hr9
2.72604
hr14
2.57481
hr19
3.15694
hr5
0.54520
hr10
0.72321
hr15
2.45019
hr20
1.51700
Tzai-Zang Lee and Ya-Fen Tseng/Asia Pacific Management Review (2006) 11(6), 389-392
Table 4. The Optimal Weight on Organizational
Performance
Organization
Performance
%
Organization
Performance
%
op1
op2
op3
0.48082
0.74243
2.35914
op5
op6
op7
0.61776
0.72210
2.70158
Charnes, A., Cooper, W. W., and Rhodes, E. (1978). Measuring the efficiency of decision-making units. European Journal of Operational
Research, 2, 429-444.
op4
Oral, M., Kettani, O. and Lang, P. (1991). A methodology for collective
0.46132
evaluation and selection of industrial R&D projects. Management
op8
Science, 37(7), 871-885.
Ruggiero, J. (2004). Data envelopment analysis with stochastic data.
91.9149
Journal of the Operational Research Society, 55, 1008-1012.
5. Conclusions
Satty, T. L. (1980). The Analytic Hierarchy Process. McGraw-Hill International Book Company.
This study uses the new method for ranking scaling
units with multiple inputs and multiple outputs in the DEA
and AHP. The advantage of the AHP/DEA ranking model
is that the AHP pariwise comparisons are got from the
input and output data, by running pairwise DEA. Thus,
there is no subjective appraisal. In addition to seeing the
efficiency value of each DMU, it can be found input and
output weight of each DMU by cross-evaluation. Moreover, besides getting the rank from the AHP/DEA, this
research can obtain the more objective weight by the cross
evaluation. It not only reveals which unit is the efficiency
but also presents which the combination of the human resource practices can obtain the performance of the unit
efficient. So, this paper clearly confers the human resource
variables and builds the model to get the optimal allocation.
Sinuany-Stern, Z., Mehrez, A. and Barboy, A. (1994). Academic departments’ efficiency in DEA. Computer and Operation Research, 21(
5), 543-556.
Sinuany-Stern, Z. and Friedman, L. (1998a). DEA and the discriminant
analysis of ratios for ranking units. European Journal of Operations
Research, 111, 470-478.
Sinuany-Stern, Z. and Friedman, L. (1998b). Rank scaling in the DEA
context. Studies in Regional and Urban Planning, 6, 135-144.
Sinuany-Stern, Z., Mehrez, A. and Hadad, Y. (2000). An AHP/DEA
methodology for ranking decision making units. International
Transactions in Operational Research, 7, 109-124.
Yang, J. and Lee, H. (1997). An AHP decision model for facility location
selection. Facilities, 15(9/10), 241-254.
Zilla, S. S., Abraham, M., Yossi, H. (2000). An AHP/DEA methodology
for ranking decision making units. International Transactions in
Operational Research 7:109-12.
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