Directory UMM :Data Elmu:jurnal:I:International Journal of Production Economics:Vol66.Issue2.Jun2000:
Int. J. Production Economics 66 (2000) 119}129
A cone-ratio DEA approach for AMT justi"cation
Srinivas Talluri*, K. Paul Yoon
Department of Information Systems and Sciences, S.J. Silberman College of Business Administration, Fairleigh Dickinson University,
H323D, 1000 River Road, Teaneck, NJ 07666, USA
Received 1 December 1998; accepted 1 July 1999
Abstract
Evaluation and selection of advanced manufacturing technology (AMT) is a complex decision making process which
requires careful consideration of various performance criteria. Initially, the decision-maker must identify a feasible set of
AMT candidates, which broadly meet the budget constraints and system requirements. The competitive priorities (cost,
time, quality, and #exibility) must then be set and matched against the performance variables in identifying preference
relationships. The "nal step involves the evaluation of alternative systems based on performance criteria and preference
relationships. This paper depicts the AMT selection process through an IDEF0 functional model. It utilizes a combination of a cone-ratio data envelopment analysis (CRDEA), which integrates decision-maker's preferences, and a new
methodological extension in data envelopment analysis (DEA). The applicability of the proposed model is illustrated by
using a real data set of industrial robots. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: AMT; DEA; Cone-ratios; Preference relationships; Relative e$ciency
1. Introduction
In order to respond quickly and e!ectively to the
rapidly changing needs of the customer and to
maintain a high level of competitiveness in the
global arena, manufacturers all around the globe
have sought the adoption of advanced manufacturing technologies (AMT) such as industrial robots,
#exible manufacturing systems (FMS), automated
material handling (AMH) systems, etc. Although it
seemed at "rst that the chief advantage of AMTs is
labor saving, it was quickly realized by many
manufacturers that AMTs result in improved prod-
* Corresponding author. Tel.: #1-201-692-7284; fax: #1201-692-7219.
E-mail address: talluri@alpha.fdu.edu (S. Talluri).
uct quality, fast production and delivery, and
increased product #exibility.
It is estimated that U.S. manufacturers have invested $32.3 billion in shop-#oor automation in
1990, and the levels of spending have increased
since then. Because of growing magnitude of investments in automation, the evaluation and selection
of AMT is a problem that requires serious consideration. The selection procedure is a complex strategic process involving many decision variables.
The decision-maker must identify the critical performance attributes, preference relationships
among the attributes, and appropriate evaluation
models to be used in the selection process.
Fig. 1 depicts the corporation wise technology
selection process through an IDEF0 functional
model [1]. IDEF0 methodology allows for a
systematic representation of steps involved in the
0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 1 2 3 - 1
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S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129
Fig. 1. Technology selection process.
decision making process, and e!ectively models the
relationships between various phases of the process. The IDEF0 model for the technology selection
process involves several activities such as identify
feasible systems, select performance variables,
identify competitive priorities, prioritize performance variables, and evaluate feasible systems. At
each stage of this process inputs are transformed
into outputs in the presence of some constraints.
For example, in the "rst activity of the technology
selection process that involves the identi"cation of
feasible systems, the inputs that include all possible
systems are evaluated in the presence of constraints
such as budget and system requirements in deriving
outputs which are the feasible systems. Similarly, in
the prioritization of the performance variables, the
inputs are the variables themselves, the constraints
are competitive priorities and strategies, and the
resulting outputs are prioritized constraints. The
decision-maker acts as a mechanism in conducting
this entire process.
Initially, the decision-maker identi"es feasible
systems from all possible systems that broadly meet
the budget constraints and system requirements of
the organization. The performance variables that
are critical to system requirements, which are utilized in AMT systems evaluation, are then selected
from a list of all system performance variables. The
next step involves the identi"cation of competitive
priorities that are in alignment with the corporate
goals and competitive strategy of the organization.
These competitive priorities are usually customer
oriented performance measures such as low cost,
high quality, short lead times, and high #exibility.
The selected variables are then matched against
these competitive priorities in identifying preference relationships among the variables. Finally, the
feasible systems are evaluated by incorporating the
prioritized constraints into the decision making
process. Based on this evaluation, the best system is
selected. If the selected system does not meet the
decision-maker's requirements for reasons beyond
S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129
the scope of this process, then new systems and/or
other performance variables are considered for the
re-evaluation process.
This paper aims at the "nal stage of the IDEF0
model, which is evaluation of feasible candidates.
The evaluation process is performed by utilizing
a combination of CRDEA and new methodological
extensions in DEA. The primary advantage of this
methodology is that it e!ectively integrates the
decision-maker's preferences on output measures.
The rest of the paper is organized as follows. Section 2 provides a literature review of the models
used in AMT evaluation. Section 3 provides an
introduction to the CRDEA model and its extensions. Section 4 presents the performance evaluation of industrial robots. Finally, the conclusions
and future study are presented in Section 5.
2. Literature review
Several models have been proposed in the literature on evaluation, justi"cation and selection of
AMT. Le#ey [2], Leung and Tanchoco [3], Proctor and Canada [4], and Sarkis [5] have presented
reviews on the evolution of justi"cation
methodologies of capital projects in general, and
manufacturing technology in particular. Proposed
methodologies have included case studies, empirical research, analytical, and simulation modeling.
In this paper, we focus on the case of AMTs.
Canada [6] developed a weighted model for
evaluating computer integrated manufacturing
(CIM) systems by incorporating "nancial measures
and certain intangible factors. Huang and Ghandforoush [7] evaluated industrial robot vendors
based on economic measures and some subjective
factors. They assigned speci"c weights to objective
and subjective factor measures in identifying the
optimal choice. However, their model did not include any technical robot performance measures.
Imany and Schlesinger [8] compared linear goal
programming (LGP) and ordinary least-squares
(OLS) methods for robot selection. The performance measures included by them are cost, load
capacity, velocity, and repeatability of the robot.
They concluded that LGP model provides more
stable results than OLS approach in the presence of
121
outliers. In addition, they stated that when several
con#icting goals need to be considered, weighted
priorities should be assigned to robot parameters.
Frazelle [9] used the analytical hierarchy process
(AHP) for evaluating material handling alternatives. According to Frazelle, a major advantage of
AHP over simple weighted evaluation techniques is
its accommodation of decision maker's inconsistencies on preference judgement. Siha [10] utilized
multi-goal programming (0}1) and AHP for robot
selection. The model included four priority levels
with various goals in each level. Equal importance
was assigned to robot performance measures such
as repeatability, reach, and cost. Maimon and
Fisher [11] developed an integer programming
model and a rule based expert system for robot
selection. Seidmann et al. [12] used a combination
of multi-goal programming and AHP for the same
problem.
Several researchers have utilized multi-attribute
decision making (MADM) models for technology
selection [13}16]. Most MADM models in AMT
evaluation assume that preferences for technology
attributes are independent of each other. But this is
not true in case of AMTs such as robots where
certain performance measures get worse as investment cost decreases. Hence, the mutual preferential
independence assumption of these models may be
violated. Khouja [17] used DEA and MADM
model together for technology selection. Although
this two-phase model is innovative, there are certain issues associated with the traditional DEA
model used in this approach. The primary limitation of this model in the context of technology
selection is that the simple radial e$ciency scores
obtained from it may not lead to the selection of the
best choice. Shang and Sueyoshi [18] utilized
a combination of AHP and DEA for selection of
#exible manufacturing systems. Their model used
AHP to restrict factor weights in DEA. However,
they could not solve the di$culty with multiple
optimum solutions.
Because of the complexity of the decision making
process involved in AMT selection, all the
aforementioned literature relied on some form of
procedure that assigns weights to various performance measures. The primary problem associated
with arbitrary weights is that they are subjective,
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S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129
and it is often a di$cult task for the decision
maker to accurately assign numbers to preferences.
Narasimhan and Vickery [19] pointed that it is a
daunting task for the decision-maker to assess weighting information as the number of performance criteria
increased. Therefore, a more robust mathematical
technique that does not demand too much and too
precise information, i.e., ordinal preferences instead
of cardinal weights, from the decision-maker can
strengthen the AMT evaluation process.
This paper depicts the AMT selection process
through an IDEF0 functional model. It proposes
a combination of a CRDEA model and a new
methodological extension in DEA, while allowing
for the incorporation of decision maker's preferences. The chief advantage of the CRDEA model is
that it does not demand exact weights from the
decision-maker. It simply requires priority order
among performance criteria.
3. Cone-ratio DEA model and extensions
DEA is a mathematical programming technique,
proposed by Charnes et al. [20], which evaluates
the relative e$ciencies of a homogeneous set of
decision making units (DMUs) in the presence of
multiple inputs and outputs. Over the years several
extensions and new models have been proposed to
improve the discriminatory power of DEA.
CRDEA is one such model that allows for integration of decision maker's preferences.
3.1. Traditional CRDEA model
Thompson et al. [21] introduced the concept of
assurance regions (ARs) in DEA. According to
them, ARs can be classi"ed into cone-ratio ARs
and non-cone-ratio ARs. Polyhedral cone-ratios
are usually expressed in intersection form or in sum
form [22]. Non-cone-ratio ARs can be expressed as
linked-cones, which link inputs and outputs, and
also as non-Archimedean and lower bound multipliers. A parallel study conducted by Charnes et al.
[22] lead to the development of the CRDEA
model. In the CRDEA model, ARs are speci"ed by
de"ning bounds on weights that re#ect the relative
importance of di!erent inputs and outputs. These
weight constraints are represented in the form of
linear inequalities and appended to the ratio DEA
model. The following formulation is the CRDEA
model with the input and output cones speci"ed by
(1.3) and (1.4), respectively:
max
s.t.
+s v y
k/1 k kp
+m u x
j/1 j jp
+s v y
k/1 k ki )1 ∀i,
+m u x
j/1 j ji
a u )u )b u , j"2, 3,2, m,
j 1
j
j 1
a v )v )b v , k"2, 3,2, s,
k 1
k
k 1
v , u *0 ∀k, j,
k j
i"1,2, n, j"1,2, m, k"1,2, s,
(1.1)
(1.2)
(1.3)
(1.4)
(1.5)
where p is the test DMU which is being evaluated,
s represents the number of outputs, m represents
the number of inputs, n is the number of decision
making units, y is the amount of output k prodki
uced by DMU i, x is the amount of input j used by
ji
DMU i, v is the weight given to output k, u is the
k
j
weight given to input j, and a, b, a, b are non-negative scalars.
The above fractional program can be converted
to the linear programming problem shown as (2):
s
max + v y ,
k kp
k/1
s.t.
m
+ u x "1,
j jp
j/1
s
m
+ v y ! + u x )0 ∀i,
k ki
j ji
k/1
j/1
a u !u )0, j"2, 3,2, m,
j 1
j
b u !u *0, j"2,3,2, m,
j 1
j
(2)
a v !v )0, k"2, 3,2, s,
k 1
k
b v !v *0, k"2, 3,2, s,
k 1
k
v , u *0 ∀k, j.
k j
This LP model is solved n times in estimating the
relative e$ciencies of all the DMUs. The model
allows each DMU to e!ectively select optimal
weights, within the speci"ed weight restrictions,
S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129
that maximize the output to input ratio, but at the
same time the constraint set prevents the e$ciencies of the n DMUs computed with these weights
from exceeding a value of `onea. A relative e$ciency score of `onea indicates that the DMU under
consideration is e$cient, whereas a score less than
`onea indicates that it is ine$cient.
The above CRDEA model has certain advantages over the ratio DEA model. The e$ciency score
obtained from the ratio DEA model is referred to as
the simple radial e$ciency. In the estimation of
simple radial e$ciency, the ratio model allows for
unrestricted factor weights (v and u ). This may
k
j
result in allowing a DMU to achieve a high relative
e$ciency score by involving in unreasonable
weighting scheme [23,24]. Such DMUs heavily
weigh few favorable measures and completely
ignore the other inputs and outputs. To overcome
this problem CRDEA model allows for weight
restrictions, which signi"cantly improve the discriminatory power of DEA.
Although the cone restrictions e!ectively discriminate between e$cient and ine$cient DMUs,
they do not allow for a ranking of e$cient units.
This poses a problem to the decision-maker in
situations where there are multiple e$cient units.
Cross-e$ciency analysis in DEA can be utilized in
such situations to discriminate between e$cient
units [25]. The cross e$ciency matrix (CEM) provides information on how well a DMU is performing with respect to the optimal DEA weights of
other DMUs. The element in the ith row and jth
column of the matrix indicates how well DMU j is
performing when evaluated with the optimal
weights of DMU i. A DMU with several high
e$ciencies along its column in the CEM is considered to be a superior performer, and a DMU
with several low e$ciencies can be regarded as
a relatively poor performer. The column means can
be computed as an index to identify the best choice.
3.2. Methodological extensions
Although the CRDEA model e!ectively integrates decision maker's preferences into the evaluation process, it does not solve the problem of
multiple optimum solutions in DEA. This problem
occurs in DEA if there are more than one set of
123
input/output weights that maximize the e$ciency
of a DMU. In such situations the usefulness of
CEM, which is derived from these weights, will be
undermined. To overcome these problems and to
provide a robust set of weights that not only maximize the e$ciency of a DMU but also minimize the
e$ciencies of its competitors we provide methodological extensions to the CRDEA model case. The
formulation shown as problem (3) is an extension
to the Doyle and Green's [26] aggressive formulation:
∀c
A
s
min + vc + y c
k c c ke
e Ep
k/1
s.t.
A
B
B
(3.1)
m
(3.2)
+ uc + x c "1,
j c c je
e Ep
j/1
m
s
(3.3)
+ vc y c ! + uc x c )0 ∀ecOpc,
j je
k ke
j/1
k/1
s
m
+ vc y c ! + uc x c "0,
(3.4)
k kp
j jp
k/1
j/1
(3.5)
ac uc !uc )0, j"2, 3,2, m,
j
j 1
(3.6)
bc uc !uc *0, j"2, 3,2, m,
j
j 1
(3.7)
ac vc !vc )0, k"2, 3,2, s,
k
k 1
(3.8)
bc vc !vc *0, k"2, 3,2, s,
k
k 1
(3.9)
vc , uc *0 ∀k, j,
k j
where c"1,2, r, and r represents the number of
cone scenarios, ec is an index that represents the
number of e$cient DMUs with respect to cone c,
and pc is the test DMU. The other notation for
inputs and outputs is same as before.
Expression (3.1) represents the output of the
composite DMU constructed from all the CRDEA
e$cient DMUs, with respect to cone c, except test
DMU pc. Expression (3.2) represents the input of
the composite DMU, and (3.3) represents the
normalization constraints. The constraint that
blankets the e$ciency score of DMU pc, which is
obtained from problem (2), is shown as (3.4).
Constraints (3.5) to (3.8) are the cone restrictions
speci"ed by the decision-maker.
124
S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129
Problem (3) identi"es weights that not only
maximize the e$ciency score of a DMU, but also
minimize the e$ciencies of all other CRDEA e$cient DMUs in some sense (by minimizing the
e$ciency of the composite DMU). We did not
include the ine$cient DMUs in problem (3), because if a system is ine$cient then it is not considered as a potential candidate for selection.
The primary contribution of the above model is
that it provides robust input/output weights for
each of the e$cient DMUs that can be utilized for
cross-e$ciency calculations to discriminate and select DMUs with best operating practices.
4. Performance evaluation of industrial robots
The robot data set published in Siha [10], shown
in Table 1, is utilized to illustrate the use of the
CRDEA model for technology selection. Thirteen
robots are considered as the feasible alternatives.
The robot performance measures considered for
the analysis are cost, repeatability, load capacity,
and velocity. These measures are not exhaustive by
any means, but frequently used in robot's performance evaluation. Repeatability can be de"ned as
the ability of the robot to return repeatedly to
a learned point in its work envelope, which is
expressed in millimeters (mm). Load capacity in
kilograms (kg) is the maximum load that the robot
Table 1
List of robots and their attributes [6]
Robot
1
2
3
4
5
6
7
8
9
10
11
12
13
can lift, which includes the actual load and the
gripper weight. Velocity of the robot is the distance
covered by the robot-arm per unit time (m/s).
In this analysis, cost is treated as an input to the
CRDEA model, because from the decision maker's
standpoint it represents the investment required to
purchase the system. Repeatability, load capacity,
and velocity are treated as outputs, because they
are measures that indicate the ability of the system
in performing various tasks.
In DEA, large value of an output is considered to
be better than small. Since smaller values of repeatability indicate better performance than larger
values, we have used inverse of repeatability in our
computations. Also, we have re-scaled the entire
data set by dividing each performance variable
value by its respective mean to remove any scale
e!ects, especially when weight restrictions are speci"ed in the CRDEA model. This is a normalization
procedure that is commonly used in DEA computations. The scaled data set is shown in Table 2.
The models discussed and proposed in this paper
are applied in a sequential manner. Initially, the
limitations of the ratio DEA model (without weight
restrictions) are demonstrated. The advantages of
the CRDEA model over the ratio DEA model are
then discussed. Finally, the results of the CRDEA
model are utilized in problem (3) in deriving robust
input/output weights, which are used for the "nal
selection process.
Table 2
Scaled robot data set
Cost
($10000)
Repeat
(mm)
Load
(kg)
Velocity
(m/s)
Robot
Cost
Repeat
Load
Velocity
6.00
9.00
9.50
12.50
7.00
7.20
4.25
5.80
11.00
5.00
4.00
8.50
9.00
0.20
0.40
0.08
1.00
0.05
1.25
0.10
0.15
0.025
0.20
0.50
1.25
0.25
6
60
28
60
20
115
50
15
10
40
10
100
70
1.50
2.90
1.00
1.20
2.50
0.90
0.70
1.50
0.67
1.00
0.50
0.90
0.65
1
2
3
4
5
6
7
8
9
10
11
12
13
0.789
1.184
1.250
1.645
0.921
0.947
0.559
0.763
1.447
0.658
0.526
1.118
1.184
0.590
0.295
1.474
0.118
2.358
0.094
1.179
0.786
4.717
0.590
0.236
0.094
0.472
0.134
1.336
0.623
1.336
0.445
2.560
1.113
0.334
0.223
0.890
0.223
2.226
1.558
1.230
2.377
0.820
0.984
2.049
0.738
0.574
1.230
0.549
0.820
0.410
0.738
0.533
S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129
4.1. Ratio DEA model results
Initially, the ratio DEA model without the
weight restrictions is used and simple radial e$ciency scores are obtained for all the robots. The
DEA formulation with robot 1 as the test DMU is
shown in Table 3. The e$ciency scores of the thirteen robots are shown under the heading EFF in
Table 4. DEA identi"ed robots 2, 5, 6, 7 and 9 to be
e$cient with a relative e$ciency score of `onea.
The other eight robots were ine$cient with e$ciency scores of less than `onea. These evaluations did
Table 3
DEA LP formulation for robot selection
Max 0.59v #0.134v #1.23v
1
2
3
s.t
0.789u "1
1
0.59v #0.134v #1.23v !0.789u )0
1
2
3
1
0.295v #1.336v #2.377v !1.184u )0
1
2
3
1
1.474v #0.623v #0.82v !1.25u )0
1
2
3
1
0.118v #1.336v #0.984v !1.645u )0
1
2
3
1
2.358v #0.445v #2.049v !0.921u )0
1
2
3
1
0.094v #2.56v #0.738v !0.947u )0
1
2
3
1
1.179v #1.113v #0.574v !0.559u )0
1
2
3
1
0.786v #0.334v #1.23v !0.763u )0
1
2
3
1
4.717v #0.223v #0.549v !1.447u )0
1
2
3
1
0.59v #0.89v #0.82v !0.658u )0
1
2
3
1
0.236v #0.223v #0.41v !0.526u )0
1
2
3
1
0.094v #2.226v #0.738v !1.118u )0
1
2
3
1
0.472v #1.558v #0.533v !1.184u )0
1
2
3
1
v , v , u , u *0
1 2 1 2
125
not include any decision-maker's preferences
(weight restrictions).
It is a di$cult task for the decision-maker to
select the optimal choice from the "ve simple radial
e$cient robots because:
(i) There may be robots indulging in an inappropriate weight scheme among the "ve e$cient
robots. Thus, an arbitrarily selected e$cient
robot may be a sub-optimal choice. Also, the
selected robot may not be the one that best
meets the decision-maker's requirements.
(ii) Alternately, the decision-maker may have to
assign speci"c weights to the attributes of the
"ve e$cient robots in selecting the optimal
choice. But from earlier discussions, it may be
a di$cult task to obtain exact weights for each
of the attributes.
Thus, using simple radial e$ciency scores alone
may not guarantee the selection of the best robot.
Hence, we need to modify the model to allow for
the decision-maker's preferences.
The competitive priorities considered in this paper are quality, lead-time, and #exibility. Quality
can be e!ectively linked to the repeatability of the
robot, because repeatability is a measure of how
well the robot performs repetitive operations. Higher
repeatability results in better consistency, which
results in higher quality of products. Similarly,
lead-time is related to the velocity of the robot.
Table 4
Simple and cone-ratio e$ciencies of robots
Robot
EFF
Cone 1
v *v *v
1
2
3
Cone 2
v *v *v
1
3
2
Cone 3
v *v *v
2
1
3
Cone 4
v *v *v
2
3
1
Cone 5
v *v *v
3
1
2
Cone 6
v *v *v
3
2
1
Mean
1
2
3
4
5
6
7
8
9
10
11
12
13
0.700
1.000
0.472
0.427
1.000
1.000
1.000
0.737
1.000
0.829
0.400
0.767
0.532
0.469
0.642
0.472
0.282
1.000
0.697
1.000
0.585
1.000
0.669
0.313
0.531
0.422
0.481
0.642
0.452
0.281
1.000
0.679
0.973
0.585
1.000
0.663
0.313
0.519
0.415
0.469
0.657
0.451
0.344
1.000
1.000
1.000
0.588
0.833
0.682
0.318
0.747
0.532
0.542
0.911
0.448
0.407
1.000
1.000
1.000
0.653
0.720
0.801
0.373
0.762
0.532
0.700
0.902
0.443
0.281
1.000
0.679
0.973
0.724
0.760
0.663
0.350
0.519
0.411
0.700
1.000
0.443
0.427
1.000
1.000
1.000
0.737
0.720
0.829
0.400
0.767
0.530
0.560
0.792
0.452
0.337
1.000
0.843
0.991
0.645
0.839
0.718
0.345
0.641
0.474
126
S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129
Higher velocity of robot-arm results in lower processing times, which decreases the overall product
lead-time. Finally, higher load capacity results in
higher load #exibility. This is because the ranges of
load capacities that can be handled by the robot are
higher thereby resulting in improved #exibility.
Based on this reasoning, we have selected six possible cone scenarios (3! cases), shown in Table 5,
where each of them can be represented as a constraint set in the CRDEA model. For example,
cone 1 represents the situation where the decision
maker sets quality, #exibility, and lead time in that
order as the competitive priorities thereby allowing
higher weight for repeatability (< ) over load capa1
city (< ), and load capacity (< ) over velocity (< ).
2
2
3
This prioritization is represented as a set of preference relationships, which are incorporated into the
CRDEA model as linear constraints. Although in
the construction of the cones we did not include
actual managerial information in terms of more
Table 5
Cones utilized in the decision making process
Cone
number
Prioritization of
performance
variables
Preference
relationships
CRDEA
constraints
1
Repeatability,
Load capacity,
Velocity
< *< *<
1
2
3
< !< *0
1
2
< !< *0
2
3
2
Repeatability,
Velocity,
Load capacity
< *< *<
1
3
2
< !< *0
1
3
< !< *0
3
2
3
Load capacity,
Repeatability,
Velocity
< *< *<
2
1
3
< !< *0
2
1
< !< *0
1
3
4
Load capacity,
Velocity,
Repeatability
< *< *<
2
3
1
< !< *0
2
3
< !< *0
3
1
Velocity,
Repeatability,
Load capacity
< *< *<
3
1
2
< !< *0
3
1
< !< *0
1
2
Velocity,
Load capacity,
Repeatability
< *< *<
3
2
1
5
6
< !< *0
3
2
< !< *0
2
1
speci"c and appropriate bounds, this information if
available can be easily integrated into the CRDEA
model. However, we did specify cones that are
e!ectively linked to the competitive priorities, and
allow for increased weights on higher priority variables.
4.2. CRDEA model results
Cone 1 analysis depicts the scenario where the
decision-maker sets the competitive priorities
thereby allowing higher weight for repeatability
(< ) over load capacity (< ), and load capacity
1
2
(< ) over velocity (< ). The results are depicted in
2
3
Table 4 under the heading `Cone 1a. Robots 5, 7,
and 9 are identi"ed to be e$cient with a relative
e$ciency score of `onea. All the other robots are
ine$cient with relative e$ciency scores of less than
`onea. Thus, the decision-maker must identify the
best robot from the three e$cient choices.
Problem (3) is utilized to identify input/output
weights for Robots 5, 7, and 9, which are used in the
evaluation of the CEM. The corresponding CEM
of the order 3]3 is depicted in Table 6. Based on
this CEM, the column means of the CEM for
Robots 5, 7, and 9 are 0.843, 0.873, and 0.851,
respectively. Robot 7 achieved the highest mean
cross-e$ciency score of 0.873. Thus, the optimal
choice for the decision-maker is Robot 7, which is
evaluated to be the best performer by the other two
e$cient robots.
Before selecting Robot 7 the decision-maker
needs to make sure that it closely satis"es other
factors such as worker approval, #oor space requirements, vendor responsiveness, etc. If the selected robot violates any of these factors
signi"cantly then the next best option must be
Table 6
CEM for Cone 1 analysis
Robot
5
7
9
5
7
9
1.000
0.742
0.785
0.973
1.000
0.647
0.720
0.833
1.000
Mean
0.843
0.873
0.851
S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129
considered. The decision-maker can identify the
next best option based on the highest cross-e$ciency mean, which in this case is Robot 9.
In cone 2 analysis the decision-maker allows
higher weight for repeatability (< ) over velocity
1
(< ), and velocity (< ) over load capacity (< ).
3
3
2
CRDEA identi"ed Robots 5 and 9 to be e$cient,
and the other eleven robots are evaluated to be
ine$cient. The cross-e$ciency means of the Robots
5 and 9 are 0.893 and 0.860, respectively. Thus, the
decision-maker must select Robot 5 as the optimal
choice.
In cone 3 analysis, we assume that the decisionmaker sets the competitive priorities thereby allowing higher weight for load capacity (< ) over
2
repeatability (< ), and repeatability (< ) over
1
1
velocity (< ). CRDEA identi"ed Robots 5, 6 and
3
7 to be e$cient, and the other ten robots are evaluated to be ine$cient. The cross-e$ciency means of
the Robots 5, 6, and 7 are 0.640, 0.788, and 0.903,
respectively. Therefore the optimal choice for the
decision-maker is Robot 7.
Cone 4 analysis allows for higher weight for load
capacity (< ) over velocity (< ), and velocity (< )
2
3
3
127
over repeatability (< ). CRDEA identi"ed Robots
1
5, 6 and 7 to be e$cient, and the other ten robots
are evaluated to be ine$cient. The cross-e$ciency
means of the Robots 5, 6, and 7 are 0.616, 0.893, and
0.903, respectively. Thus, the decision-maker must
select Robot 7 as the optimal choice.
Cone 5 analysis illustrates the situation where
the decision-maker allows higher weight for velocity (< ) over repeatability (< ), and repeatability
3
1
(< ) over load capacity (< ). In this analysis only
1
2
Robot 5 is identi"ed to be e$cient. Since no other
robot is e$cient, there is no need for cross-e$ciency analysis. The decision-maker can select Robot
5 as the optimal choice.
Finally, in cone 6 analysis the decision-maker
allows higher weight for velocity (< ) over load
3
capacity (< ), and load capacity (< ) over repeata2
2
bility (< ). CRDEA identi"ed Robots 2, 5, 6, and
1
7 to be e$cient. The mean cross-e$ciency means of
Robots 2, 5, 6, and 7 are 0.886, 0.944, 0.713, and
0.760, respectively. So, the decision-maker can select Robot 5 as the optimal choice.
In summary, the selected choice is dependent on
the competitive priorities set. This analysis allows
Fig. 2. Comparison of robot e$ciencies.
128
S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129
the decision-maker to identify the optimal choice
based on the critical performance variables and
system requirements that are vital for satisfying the
corporate goals and long-term strategy of the organization. If the decision maker is interested in
a good overall performer then they have to go with
Robot 5, which performed consistently well across
all cones and achieved a mean score (mean of all
six cone e$ciencies) of 1.000, shown in Table 4
under the heading `Meana. Fig. 2 helps facilitate
the comparison of the 13 robots across all the six
cone scenarios. It clearly demonstrates the variability and consistency in the performance of the robots with respect to the decision-maker's
preferences.
5. Conclusions and future study
In this paper a CRDEA model is proposed for
AMT evaluation and selection. The strength of the
model is in incorporating the decision-maker's input and e!ectively discriminating between e$cient
and ine$cient technologies. The limitations of the
ratio DEA model are illustrated, and its results are
compared to that of the CRDEA model. The paper
also presents a new methodological extension in
DEA to improve the discriminatory power of the
CRDEA model.
As an extension to this study, it will be interesting to investigate the impact of the linked-cone
structures, linking inputs and outputs, on the decision making process. However, the decision-maker
must be cautious when using these linked-cone
structures because of possible infeasibility problems. For more information on application of linked-cones, see Thompson et al. [27].
One of the limitations of this study is that the
cone restrictions are not derived from actual managerial experiences, but we did provide cones that
are e!ectively linked to the competitive priorities.
Although it may be a di$cult task to obtain cone
ratios from actual managerial practice, it is relatively easier to obtain these types of bounds than
specifying exact weights. We would like to point
out that this model can easily integrate such
bounds into the decision making process, if available. Another limitation of this research is that the
model does not allow for the incorporation of
qualitative factors into the selection process.
Although DEA was applied extensively in evaluating the e$ciencies of hospitals, schools, banks,
health insurance, and other pro"t and non-pro"t
organizations, its application in production and
operations "eld has been very limited. This paper
illustrates its importance as a decision making tool
in the area of technology management.
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A cone-ratio DEA approach for AMT justi"cation
Srinivas Talluri*, K. Paul Yoon
Department of Information Systems and Sciences, S.J. Silberman College of Business Administration, Fairleigh Dickinson University,
H323D, 1000 River Road, Teaneck, NJ 07666, USA
Received 1 December 1998; accepted 1 July 1999
Abstract
Evaluation and selection of advanced manufacturing technology (AMT) is a complex decision making process which
requires careful consideration of various performance criteria. Initially, the decision-maker must identify a feasible set of
AMT candidates, which broadly meet the budget constraints and system requirements. The competitive priorities (cost,
time, quality, and #exibility) must then be set and matched against the performance variables in identifying preference
relationships. The "nal step involves the evaluation of alternative systems based on performance criteria and preference
relationships. This paper depicts the AMT selection process through an IDEF0 functional model. It utilizes a combination of a cone-ratio data envelopment analysis (CRDEA), which integrates decision-maker's preferences, and a new
methodological extension in data envelopment analysis (DEA). The applicability of the proposed model is illustrated by
using a real data set of industrial robots. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: AMT; DEA; Cone-ratios; Preference relationships; Relative e$ciency
1. Introduction
In order to respond quickly and e!ectively to the
rapidly changing needs of the customer and to
maintain a high level of competitiveness in the
global arena, manufacturers all around the globe
have sought the adoption of advanced manufacturing technologies (AMT) such as industrial robots,
#exible manufacturing systems (FMS), automated
material handling (AMH) systems, etc. Although it
seemed at "rst that the chief advantage of AMTs is
labor saving, it was quickly realized by many
manufacturers that AMTs result in improved prod-
* Corresponding author. Tel.: #1-201-692-7284; fax: #1201-692-7219.
E-mail address: talluri@alpha.fdu.edu (S. Talluri).
uct quality, fast production and delivery, and
increased product #exibility.
It is estimated that U.S. manufacturers have invested $32.3 billion in shop-#oor automation in
1990, and the levels of spending have increased
since then. Because of growing magnitude of investments in automation, the evaluation and selection
of AMT is a problem that requires serious consideration. The selection procedure is a complex strategic process involving many decision variables.
The decision-maker must identify the critical performance attributes, preference relationships
among the attributes, and appropriate evaluation
models to be used in the selection process.
Fig. 1 depicts the corporation wise technology
selection process through an IDEF0 functional
model [1]. IDEF0 methodology allows for a
systematic representation of steps involved in the
0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 1 2 3 - 1
120
S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129
Fig. 1. Technology selection process.
decision making process, and e!ectively models the
relationships between various phases of the process. The IDEF0 model for the technology selection
process involves several activities such as identify
feasible systems, select performance variables,
identify competitive priorities, prioritize performance variables, and evaluate feasible systems. At
each stage of this process inputs are transformed
into outputs in the presence of some constraints.
For example, in the "rst activity of the technology
selection process that involves the identi"cation of
feasible systems, the inputs that include all possible
systems are evaluated in the presence of constraints
such as budget and system requirements in deriving
outputs which are the feasible systems. Similarly, in
the prioritization of the performance variables, the
inputs are the variables themselves, the constraints
are competitive priorities and strategies, and the
resulting outputs are prioritized constraints. The
decision-maker acts as a mechanism in conducting
this entire process.
Initially, the decision-maker identi"es feasible
systems from all possible systems that broadly meet
the budget constraints and system requirements of
the organization. The performance variables that
are critical to system requirements, which are utilized in AMT systems evaluation, are then selected
from a list of all system performance variables. The
next step involves the identi"cation of competitive
priorities that are in alignment with the corporate
goals and competitive strategy of the organization.
These competitive priorities are usually customer
oriented performance measures such as low cost,
high quality, short lead times, and high #exibility.
The selected variables are then matched against
these competitive priorities in identifying preference relationships among the variables. Finally, the
feasible systems are evaluated by incorporating the
prioritized constraints into the decision making
process. Based on this evaluation, the best system is
selected. If the selected system does not meet the
decision-maker's requirements for reasons beyond
S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129
the scope of this process, then new systems and/or
other performance variables are considered for the
re-evaluation process.
This paper aims at the "nal stage of the IDEF0
model, which is evaluation of feasible candidates.
The evaluation process is performed by utilizing
a combination of CRDEA and new methodological
extensions in DEA. The primary advantage of this
methodology is that it e!ectively integrates the
decision-maker's preferences on output measures.
The rest of the paper is organized as follows. Section 2 provides a literature review of the models
used in AMT evaluation. Section 3 provides an
introduction to the CRDEA model and its extensions. Section 4 presents the performance evaluation of industrial robots. Finally, the conclusions
and future study are presented in Section 5.
2. Literature review
Several models have been proposed in the literature on evaluation, justi"cation and selection of
AMT. Le#ey [2], Leung and Tanchoco [3], Proctor and Canada [4], and Sarkis [5] have presented
reviews on the evolution of justi"cation
methodologies of capital projects in general, and
manufacturing technology in particular. Proposed
methodologies have included case studies, empirical research, analytical, and simulation modeling.
In this paper, we focus on the case of AMTs.
Canada [6] developed a weighted model for
evaluating computer integrated manufacturing
(CIM) systems by incorporating "nancial measures
and certain intangible factors. Huang and Ghandforoush [7] evaluated industrial robot vendors
based on economic measures and some subjective
factors. They assigned speci"c weights to objective
and subjective factor measures in identifying the
optimal choice. However, their model did not include any technical robot performance measures.
Imany and Schlesinger [8] compared linear goal
programming (LGP) and ordinary least-squares
(OLS) methods for robot selection. The performance measures included by them are cost, load
capacity, velocity, and repeatability of the robot.
They concluded that LGP model provides more
stable results than OLS approach in the presence of
121
outliers. In addition, they stated that when several
con#icting goals need to be considered, weighted
priorities should be assigned to robot parameters.
Frazelle [9] used the analytical hierarchy process
(AHP) for evaluating material handling alternatives. According to Frazelle, a major advantage of
AHP over simple weighted evaluation techniques is
its accommodation of decision maker's inconsistencies on preference judgement. Siha [10] utilized
multi-goal programming (0}1) and AHP for robot
selection. The model included four priority levels
with various goals in each level. Equal importance
was assigned to robot performance measures such
as repeatability, reach, and cost. Maimon and
Fisher [11] developed an integer programming
model and a rule based expert system for robot
selection. Seidmann et al. [12] used a combination
of multi-goal programming and AHP for the same
problem.
Several researchers have utilized multi-attribute
decision making (MADM) models for technology
selection [13}16]. Most MADM models in AMT
evaluation assume that preferences for technology
attributes are independent of each other. But this is
not true in case of AMTs such as robots where
certain performance measures get worse as investment cost decreases. Hence, the mutual preferential
independence assumption of these models may be
violated. Khouja [17] used DEA and MADM
model together for technology selection. Although
this two-phase model is innovative, there are certain issues associated with the traditional DEA
model used in this approach. The primary limitation of this model in the context of technology
selection is that the simple radial e$ciency scores
obtained from it may not lead to the selection of the
best choice. Shang and Sueyoshi [18] utilized
a combination of AHP and DEA for selection of
#exible manufacturing systems. Their model used
AHP to restrict factor weights in DEA. However,
they could not solve the di$culty with multiple
optimum solutions.
Because of the complexity of the decision making
process involved in AMT selection, all the
aforementioned literature relied on some form of
procedure that assigns weights to various performance measures. The primary problem associated
with arbitrary weights is that they are subjective,
122
S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129
and it is often a di$cult task for the decision
maker to accurately assign numbers to preferences.
Narasimhan and Vickery [19] pointed that it is a
daunting task for the decision-maker to assess weighting information as the number of performance criteria
increased. Therefore, a more robust mathematical
technique that does not demand too much and too
precise information, i.e., ordinal preferences instead
of cardinal weights, from the decision-maker can
strengthen the AMT evaluation process.
This paper depicts the AMT selection process
through an IDEF0 functional model. It proposes
a combination of a CRDEA model and a new
methodological extension in DEA, while allowing
for the incorporation of decision maker's preferences. The chief advantage of the CRDEA model is
that it does not demand exact weights from the
decision-maker. It simply requires priority order
among performance criteria.
3. Cone-ratio DEA model and extensions
DEA is a mathematical programming technique,
proposed by Charnes et al. [20], which evaluates
the relative e$ciencies of a homogeneous set of
decision making units (DMUs) in the presence of
multiple inputs and outputs. Over the years several
extensions and new models have been proposed to
improve the discriminatory power of DEA.
CRDEA is one such model that allows for integration of decision maker's preferences.
3.1. Traditional CRDEA model
Thompson et al. [21] introduced the concept of
assurance regions (ARs) in DEA. According to
them, ARs can be classi"ed into cone-ratio ARs
and non-cone-ratio ARs. Polyhedral cone-ratios
are usually expressed in intersection form or in sum
form [22]. Non-cone-ratio ARs can be expressed as
linked-cones, which link inputs and outputs, and
also as non-Archimedean and lower bound multipliers. A parallel study conducted by Charnes et al.
[22] lead to the development of the CRDEA
model. In the CRDEA model, ARs are speci"ed by
de"ning bounds on weights that re#ect the relative
importance of di!erent inputs and outputs. These
weight constraints are represented in the form of
linear inequalities and appended to the ratio DEA
model. The following formulation is the CRDEA
model with the input and output cones speci"ed by
(1.3) and (1.4), respectively:
max
s.t.
+s v y
k/1 k kp
+m u x
j/1 j jp
+s v y
k/1 k ki )1 ∀i,
+m u x
j/1 j ji
a u )u )b u , j"2, 3,2, m,
j 1
j
j 1
a v )v )b v , k"2, 3,2, s,
k 1
k
k 1
v , u *0 ∀k, j,
k j
i"1,2, n, j"1,2, m, k"1,2, s,
(1.1)
(1.2)
(1.3)
(1.4)
(1.5)
where p is the test DMU which is being evaluated,
s represents the number of outputs, m represents
the number of inputs, n is the number of decision
making units, y is the amount of output k prodki
uced by DMU i, x is the amount of input j used by
ji
DMU i, v is the weight given to output k, u is the
k
j
weight given to input j, and a, b, a, b are non-negative scalars.
The above fractional program can be converted
to the linear programming problem shown as (2):
s
max + v y ,
k kp
k/1
s.t.
m
+ u x "1,
j jp
j/1
s
m
+ v y ! + u x )0 ∀i,
k ki
j ji
k/1
j/1
a u !u )0, j"2, 3,2, m,
j 1
j
b u !u *0, j"2,3,2, m,
j 1
j
(2)
a v !v )0, k"2, 3,2, s,
k 1
k
b v !v *0, k"2, 3,2, s,
k 1
k
v , u *0 ∀k, j.
k j
This LP model is solved n times in estimating the
relative e$ciencies of all the DMUs. The model
allows each DMU to e!ectively select optimal
weights, within the speci"ed weight restrictions,
S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129
that maximize the output to input ratio, but at the
same time the constraint set prevents the e$ciencies of the n DMUs computed with these weights
from exceeding a value of `onea. A relative e$ciency score of `onea indicates that the DMU under
consideration is e$cient, whereas a score less than
`onea indicates that it is ine$cient.
The above CRDEA model has certain advantages over the ratio DEA model. The e$ciency score
obtained from the ratio DEA model is referred to as
the simple radial e$ciency. In the estimation of
simple radial e$ciency, the ratio model allows for
unrestricted factor weights (v and u ). This may
k
j
result in allowing a DMU to achieve a high relative
e$ciency score by involving in unreasonable
weighting scheme [23,24]. Such DMUs heavily
weigh few favorable measures and completely
ignore the other inputs and outputs. To overcome
this problem CRDEA model allows for weight
restrictions, which signi"cantly improve the discriminatory power of DEA.
Although the cone restrictions e!ectively discriminate between e$cient and ine$cient DMUs,
they do not allow for a ranking of e$cient units.
This poses a problem to the decision-maker in
situations where there are multiple e$cient units.
Cross-e$ciency analysis in DEA can be utilized in
such situations to discriminate between e$cient
units [25]. The cross e$ciency matrix (CEM) provides information on how well a DMU is performing with respect to the optimal DEA weights of
other DMUs. The element in the ith row and jth
column of the matrix indicates how well DMU j is
performing when evaluated with the optimal
weights of DMU i. A DMU with several high
e$ciencies along its column in the CEM is considered to be a superior performer, and a DMU
with several low e$ciencies can be regarded as
a relatively poor performer. The column means can
be computed as an index to identify the best choice.
3.2. Methodological extensions
Although the CRDEA model e!ectively integrates decision maker's preferences into the evaluation process, it does not solve the problem of
multiple optimum solutions in DEA. This problem
occurs in DEA if there are more than one set of
123
input/output weights that maximize the e$ciency
of a DMU. In such situations the usefulness of
CEM, which is derived from these weights, will be
undermined. To overcome these problems and to
provide a robust set of weights that not only maximize the e$ciency of a DMU but also minimize the
e$ciencies of its competitors we provide methodological extensions to the CRDEA model case. The
formulation shown as problem (3) is an extension
to the Doyle and Green's [26] aggressive formulation:
∀c
A
s
min + vc + y c
k c c ke
e Ep
k/1
s.t.
A
B
B
(3.1)
m
(3.2)
+ uc + x c "1,
j c c je
e Ep
j/1
m
s
(3.3)
+ vc y c ! + uc x c )0 ∀ecOpc,
j je
k ke
j/1
k/1
s
m
+ vc y c ! + uc x c "0,
(3.4)
k kp
j jp
k/1
j/1
(3.5)
ac uc !uc )0, j"2, 3,2, m,
j
j 1
(3.6)
bc uc !uc *0, j"2, 3,2, m,
j
j 1
(3.7)
ac vc !vc )0, k"2, 3,2, s,
k
k 1
(3.8)
bc vc !vc *0, k"2, 3,2, s,
k
k 1
(3.9)
vc , uc *0 ∀k, j,
k j
where c"1,2, r, and r represents the number of
cone scenarios, ec is an index that represents the
number of e$cient DMUs with respect to cone c,
and pc is the test DMU. The other notation for
inputs and outputs is same as before.
Expression (3.1) represents the output of the
composite DMU constructed from all the CRDEA
e$cient DMUs, with respect to cone c, except test
DMU pc. Expression (3.2) represents the input of
the composite DMU, and (3.3) represents the
normalization constraints. The constraint that
blankets the e$ciency score of DMU pc, which is
obtained from problem (2), is shown as (3.4).
Constraints (3.5) to (3.8) are the cone restrictions
speci"ed by the decision-maker.
124
S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129
Problem (3) identi"es weights that not only
maximize the e$ciency score of a DMU, but also
minimize the e$ciencies of all other CRDEA e$cient DMUs in some sense (by minimizing the
e$ciency of the composite DMU). We did not
include the ine$cient DMUs in problem (3), because if a system is ine$cient then it is not considered as a potential candidate for selection.
The primary contribution of the above model is
that it provides robust input/output weights for
each of the e$cient DMUs that can be utilized for
cross-e$ciency calculations to discriminate and select DMUs with best operating practices.
4. Performance evaluation of industrial robots
The robot data set published in Siha [10], shown
in Table 1, is utilized to illustrate the use of the
CRDEA model for technology selection. Thirteen
robots are considered as the feasible alternatives.
The robot performance measures considered for
the analysis are cost, repeatability, load capacity,
and velocity. These measures are not exhaustive by
any means, but frequently used in robot's performance evaluation. Repeatability can be de"ned as
the ability of the robot to return repeatedly to
a learned point in its work envelope, which is
expressed in millimeters (mm). Load capacity in
kilograms (kg) is the maximum load that the robot
Table 1
List of robots and their attributes [6]
Robot
1
2
3
4
5
6
7
8
9
10
11
12
13
can lift, which includes the actual load and the
gripper weight. Velocity of the robot is the distance
covered by the robot-arm per unit time (m/s).
In this analysis, cost is treated as an input to the
CRDEA model, because from the decision maker's
standpoint it represents the investment required to
purchase the system. Repeatability, load capacity,
and velocity are treated as outputs, because they
are measures that indicate the ability of the system
in performing various tasks.
In DEA, large value of an output is considered to
be better than small. Since smaller values of repeatability indicate better performance than larger
values, we have used inverse of repeatability in our
computations. Also, we have re-scaled the entire
data set by dividing each performance variable
value by its respective mean to remove any scale
e!ects, especially when weight restrictions are speci"ed in the CRDEA model. This is a normalization
procedure that is commonly used in DEA computations. The scaled data set is shown in Table 2.
The models discussed and proposed in this paper
are applied in a sequential manner. Initially, the
limitations of the ratio DEA model (without weight
restrictions) are demonstrated. The advantages of
the CRDEA model over the ratio DEA model are
then discussed. Finally, the results of the CRDEA
model are utilized in problem (3) in deriving robust
input/output weights, which are used for the "nal
selection process.
Table 2
Scaled robot data set
Cost
($10000)
Repeat
(mm)
Load
(kg)
Velocity
(m/s)
Robot
Cost
Repeat
Load
Velocity
6.00
9.00
9.50
12.50
7.00
7.20
4.25
5.80
11.00
5.00
4.00
8.50
9.00
0.20
0.40
0.08
1.00
0.05
1.25
0.10
0.15
0.025
0.20
0.50
1.25
0.25
6
60
28
60
20
115
50
15
10
40
10
100
70
1.50
2.90
1.00
1.20
2.50
0.90
0.70
1.50
0.67
1.00
0.50
0.90
0.65
1
2
3
4
5
6
7
8
9
10
11
12
13
0.789
1.184
1.250
1.645
0.921
0.947
0.559
0.763
1.447
0.658
0.526
1.118
1.184
0.590
0.295
1.474
0.118
2.358
0.094
1.179
0.786
4.717
0.590
0.236
0.094
0.472
0.134
1.336
0.623
1.336
0.445
2.560
1.113
0.334
0.223
0.890
0.223
2.226
1.558
1.230
2.377
0.820
0.984
2.049
0.738
0.574
1.230
0.549
0.820
0.410
0.738
0.533
S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129
4.1. Ratio DEA model results
Initially, the ratio DEA model without the
weight restrictions is used and simple radial e$ciency scores are obtained for all the robots. The
DEA formulation with robot 1 as the test DMU is
shown in Table 3. The e$ciency scores of the thirteen robots are shown under the heading EFF in
Table 4. DEA identi"ed robots 2, 5, 6, 7 and 9 to be
e$cient with a relative e$ciency score of `onea.
The other eight robots were ine$cient with e$ciency scores of less than `onea. These evaluations did
Table 3
DEA LP formulation for robot selection
Max 0.59v #0.134v #1.23v
1
2
3
s.t
0.789u "1
1
0.59v #0.134v #1.23v !0.789u )0
1
2
3
1
0.295v #1.336v #2.377v !1.184u )0
1
2
3
1
1.474v #0.623v #0.82v !1.25u )0
1
2
3
1
0.118v #1.336v #0.984v !1.645u )0
1
2
3
1
2.358v #0.445v #2.049v !0.921u )0
1
2
3
1
0.094v #2.56v #0.738v !0.947u )0
1
2
3
1
1.179v #1.113v #0.574v !0.559u )0
1
2
3
1
0.786v #0.334v #1.23v !0.763u )0
1
2
3
1
4.717v #0.223v #0.549v !1.447u )0
1
2
3
1
0.59v #0.89v #0.82v !0.658u )0
1
2
3
1
0.236v #0.223v #0.41v !0.526u )0
1
2
3
1
0.094v #2.226v #0.738v !1.118u )0
1
2
3
1
0.472v #1.558v #0.533v !1.184u )0
1
2
3
1
v , v , u , u *0
1 2 1 2
125
not include any decision-maker's preferences
(weight restrictions).
It is a di$cult task for the decision-maker to
select the optimal choice from the "ve simple radial
e$cient robots because:
(i) There may be robots indulging in an inappropriate weight scheme among the "ve e$cient
robots. Thus, an arbitrarily selected e$cient
robot may be a sub-optimal choice. Also, the
selected robot may not be the one that best
meets the decision-maker's requirements.
(ii) Alternately, the decision-maker may have to
assign speci"c weights to the attributes of the
"ve e$cient robots in selecting the optimal
choice. But from earlier discussions, it may be
a di$cult task to obtain exact weights for each
of the attributes.
Thus, using simple radial e$ciency scores alone
may not guarantee the selection of the best robot.
Hence, we need to modify the model to allow for
the decision-maker's preferences.
The competitive priorities considered in this paper are quality, lead-time, and #exibility. Quality
can be e!ectively linked to the repeatability of the
robot, because repeatability is a measure of how
well the robot performs repetitive operations. Higher
repeatability results in better consistency, which
results in higher quality of products. Similarly,
lead-time is related to the velocity of the robot.
Table 4
Simple and cone-ratio e$ciencies of robots
Robot
EFF
Cone 1
v *v *v
1
2
3
Cone 2
v *v *v
1
3
2
Cone 3
v *v *v
2
1
3
Cone 4
v *v *v
2
3
1
Cone 5
v *v *v
3
1
2
Cone 6
v *v *v
3
2
1
Mean
1
2
3
4
5
6
7
8
9
10
11
12
13
0.700
1.000
0.472
0.427
1.000
1.000
1.000
0.737
1.000
0.829
0.400
0.767
0.532
0.469
0.642
0.472
0.282
1.000
0.697
1.000
0.585
1.000
0.669
0.313
0.531
0.422
0.481
0.642
0.452
0.281
1.000
0.679
0.973
0.585
1.000
0.663
0.313
0.519
0.415
0.469
0.657
0.451
0.344
1.000
1.000
1.000
0.588
0.833
0.682
0.318
0.747
0.532
0.542
0.911
0.448
0.407
1.000
1.000
1.000
0.653
0.720
0.801
0.373
0.762
0.532
0.700
0.902
0.443
0.281
1.000
0.679
0.973
0.724
0.760
0.663
0.350
0.519
0.411
0.700
1.000
0.443
0.427
1.000
1.000
1.000
0.737
0.720
0.829
0.400
0.767
0.530
0.560
0.792
0.452
0.337
1.000
0.843
0.991
0.645
0.839
0.718
0.345
0.641
0.474
126
S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129
Higher velocity of robot-arm results in lower processing times, which decreases the overall product
lead-time. Finally, higher load capacity results in
higher load #exibility. This is because the ranges of
load capacities that can be handled by the robot are
higher thereby resulting in improved #exibility.
Based on this reasoning, we have selected six possible cone scenarios (3! cases), shown in Table 5,
where each of them can be represented as a constraint set in the CRDEA model. For example,
cone 1 represents the situation where the decision
maker sets quality, #exibility, and lead time in that
order as the competitive priorities thereby allowing
higher weight for repeatability (< ) over load capa1
city (< ), and load capacity (< ) over velocity (< ).
2
2
3
This prioritization is represented as a set of preference relationships, which are incorporated into the
CRDEA model as linear constraints. Although in
the construction of the cones we did not include
actual managerial information in terms of more
Table 5
Cones utilized in the decision making process
Cone
number
Prioritization of
performance
variables
Preference
relationships
CRDEA
constraints
1
Repeatability,
Load capacity,
Velocity
< *< *<
1
2
3
< !< *0
1
2
< !< *0
2
3
2
Repeatability,
Velocity,
Load capacity
< *< *<
1
3
2
< !< *0
1
3
< !< *0
3
2
3
Load capacity,
Repeatability,
Velocity
< *< *<
2
1
3
< !< *0
2
1
< !< *0
1
3
4
Load capacity,
Velocity,
Repeatability
< *< *<
2
3
1
< !< *0
2
3
< !< *0
3
1
Velocity,
Repeatability,
Load capacity
< *< *<
3
1
2
< !< *0
3
1
< !< *0
1
2
Velocity,
Load capacity,
Repeatability
< *< *<
3
2
1
5
6
< !< *0
3
2
< !< *0
2
1
speci"c and appropriate bounds, this information if
available can be easily integrated into the CRDEA
model. However, we did specify cones that are
e!ectively linked to the competitive priorities, and
allow for increased weights on higher priority variables.
4.2. CRDEA model results
Cone 1 analysis depicts the scenario where the
decision-maker sets the competitive priorities
thereby allowing higher weight for repeatability
(< ) over load capacity (< ), and load capacity
1
2
(< ) over velocity (< ). The results are depicted in
2
3
Table 4 under the heading `Cone 1a. Robots 5, 7,
and 9 are identi"ed to be e$cient with a relative
e$ciency score of `onea. All the other robots are
ine$cient with relative e$ciency scores of less than
`onea. Thus, the decision-maker must identify the
best robot from the three e$cient choices.
Problem (3) is utilized to identify input/output
weights for Robots 5, 7, and 9, which are used in the
evaluation of the CEM. The corresponding CEM
of the order 3]3 is depicted in Table 6. Based on
this CEM, the column means of the CEM for
Robots 5, 7, and 9 are 0.843, 0.873, and 0.851,
respectively. Robot 7 achieved the highest mean
cross-e$ciency score of 0.873. Thus, the optimal
choice for the decision-maker is Robot 7, which is
evaluated to be the best performer by the other two
e$cient robots.
Before selecting Robot 7 the decision-maker
needs to make sure that it closely satis"es other
factors such as worker approval, #oor space requirements, vendor responsiveness, etc. If the selected robot violates any of these factors
signi"cantly then the next best option must be
Table 6
CEM for Cone 1 analysis
Robot
5
7
9
5
7
9
1.000
0.742
0.785
0.973
1.000
0.647
0.720
0.833
1.000
Mean
0.843
0.873
0.851
S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129
considered. The decision-maker can identify the
next best option based on the highest cross-e$ciency mean, which in this case is Robot 9.
In cone 2 analysis the decision-maker allows
higher weight for repeatability (< ) over velocity
1
(< ), and velocity (< ) over load capacity (< ).
3
3
2
CRDEA identi"ed Robots 5 and 9 to be e$cient,
and the other eleven robots are evaluated to be
ine$cient. The cross-e$ciency means of the Robots
5 and 9 are 0.893 and 0.860, respectively. Thus, the
decision-maker must select Robot 5 as the optimal
choice.
In cone 3 analysis, we assume that the decisionmaker sets the competitive priorities thereby allowing higher weight for load capacity (< ) over
2
repeatability (< ), and repeatability (< ) over
1
1
velocity (< ). CRDEA identi"ed Robots 5, 6 and
3
7 to be e$cient, and the other ten robots are evaluated to be ine$cient. The cross-e$ciency means of
the Robots 5, 6, and 7 are 0.640, 0.788, and 0.903,
respectively. Therefore the optimal choice for the
decision-maker is Robot 7.
Cone 4 analysis allows for higher weight for load
capacity (< ) over velocity (< ), and velocity (< )
2
3
3
127
over repeatability (< ). CRDEA identi"ed Robots
1
5, 6 and 7 to be e$cient, and the other ten robots
are evaluated to be ine$cient. The cross-e$ciency
means of the Robots 5, 6, and 7 are 0.616, 0.893, and
0.903, respectively. Thus, the decision-maker must
select Robot 7 as the optimal choice.
Cone 5 analysis illustrates the situation where
the decision-maker allows higher weight for velocity (< ) over repeatability (< ), and repeatability
3
1
(< ) over load capacity (< ). In this analysis only
1
2
Robot 5 is identi"ed to be e$cient. Since no other
robot is e$cient, there is no need for cross-e$ciency analysis. The decision-maker can select Robot
5 as the optimal choice.
Finally, in cone 6 analysis the decision-maker
allows higher weight for velocity (< ) over load
3
capacity (< ), and load capacity (< ) over repeata2
2
bility (< ). CRDEA identi"ed Robots 2, 5, 6, and
1
7 to be e$cient. The mean cross-e$ciency means of
Robots 2, 5, 6, and 7 are 0.886, 0.944, 0.713, and
0.760, respectively. So, the decision-maker can select Robot 5 as the optimal choice.
In summary, the selected choice is dependent on
the competitive priorities set. This analysis allows
Fig. 2. Comparison of robot e$ciencies.
128
S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129
the decision-maker to identify the optimal choice
based on the critical performance variables and
system requirements that are vital for satisfying the
corporate goals and long-term strategy of the organization. If the decision maker is interested in
a good overall performer then they have to go with
Robot 5, which performed consistently well across
all cones and achieved a mean score (mean of all
six cone e$ciencies) of 1.000, shown in Table 4
under the heading `Meana. Fig. 2 helps facilitate
the comparison of the 13 robots across all the six
cone scenarios. It clearly demonstrates the variability and consistency in the performance of the robots with respect to the decision-maker's
preferences.
5. Conclusions and future study
In this paper a CRDEA model is proposed for
AMT evaluation and selection. The strength of the
model is in incorporating the decision-maker's input and e!ectively discriminating between e$cient
and ine$cient technologies. The limitations of the
ratio DEA model are illustrated, and its results are
compared to that of the CRDEA model. The paper
also presents a new methodological extension in
DEA to improve the discriminatory power of the
CRDEA model.
As an extension to this study, it will be interesting to investigate the impact of the linked-cone
structures, linking inputs and outputs, on the decision making process. However, the decision-maker
must be cautious when using these linked-cone
structures because of possible infeasibility problems. For more information on application of linked-cones, see Thompson et al. [27].
One of the limitations of this study is that the
cone restrictions are not derived from actual managerial experiences, but we did provide cones that
are e!ectively linked to the competitive priorities.
Although it may be a di$cult task to obtain cone
ratios from actual managerial practice, it is relatively easier to obtain these types of bounds than
specifying exact weights. We would like to point
out that this model can easily integrate such
bounds into the decision making process, if available. Another limitation of this research is that the
model does not allow for the incorporation of
qualitative factors into the selection process.
Although DEA was applied extensively in evaluating the e$ciencies of hospitals, schools, banks,
health insurance, and other pro"t and non-pro"t
organizations, its application in production and
operations "eld has been very limited. This paper
illustrates its importance as a decision making tool
in the area of technology management.
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