A cone-ratio DEA approach for AMT justification

*Corresponding author. Tel.: #1-201-692-7284; fax: #1-201-692-7219.

E-mail address: [email protected] (S. Talluri).

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 whichrequires careful consideration of various performance criteria. Initially, the decision-maker must identify a feasible set ofAMT 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 preferencerelationships. The "nal step involves the evaluation of alternative systems based on performance criteria and preferencerelationships. This paper depicts the AMT selection process through an IDEF0 functional model. It utilizes a combina-tion of a cone-ratio data envelopment analysis (CRDEA), which integrates decision-maker's preferences, and a newmethodological extension in data envelopment analysis (DEA). The applicability of the proposed model is illustrated byusing 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 therapidly changing needs of the customer and tomaintain a high level of competitiveness in theglobal arena, manufacturers all around the globehave sought the adoption of advanced manufactur-ing technologies (AMT) such as industrial robots,#exible manufacturing systems (FMS), automatedmaterial handling (AMH) systems, etc. Although itseemed at "rst that the chief advantage of AMTs islabor saving, it was quickly realized by manymanufacturers that AMTs result in improved prod-

uct quality, fast production and delivery, andincreased product #exibility.

It is estimated that U.S. manufacturers have in-vested $32.3 billion in shop-#oor automation in1990, and the levels of spending have increasedsince then. Because of growing magnitude of invest-ments in automation, the evaluation and selectionof AMT is a problem that requires serious consid-eration. The selection procedure is a complex stra-tegic process involving many decision variables.The decision-maker must identify the critical per-formance attributes, preference relationshipsamong the attributes, and appropriate evaluationmodels to be used in the selection process.

Fig. 1 depicts the corporation wise technologyselection process through an IDEF0 functionalmodel [1]. IDEF0 methodology allows for asystematic 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

Fig. 1. Technology selection process.

decision making process, and e!ectively models therelationships between various phases of the pro-cess. The IDEF0 model for the technology selectionprocess involves several activities such as identifyfeasible systems, select performance variables,identify competitive priorities, prioritize perfor-mance variables, and evaluate feasible systems. Ateach stage of this process inputs are transformedinto outputs in the presence of some constraints.For example, in the "rst activity of the technologyselection process that involves the identi"cation offeasible systems, the inputs that include all possiblesystems are evaluated in the presence of constraintssuch as budget and system requirements in derivingoutputs which are the feasible systems. Similarly, inthe prioritization of the performance variables, theinputs are the variables themselves, the constraintsare competitive priorities and strategies, and theresulting outputs are prioritized constraints. Thedecision-maker acts as a mechanism in conductingthis entire process.

Initially, the decision-maker identi"es feasiblesystems from all possible systems that broadly meetthe budget constraints and system requirements ofthe organization. The performance variables thatare critical to system requirements, which are utiliz-ed in AMT systems evaluation, are then selectedfrom a list of all system performance variables. Thenext step involves the identi"cation of competitivepriorities that are in alignment with the corporategoals and competitive strategy of the organization.These competitive priorities are usually customeroriented performance measures such as low cost,high quality, short lead times, and high #exibility.The selected variables are then matched againstthese competitive priorities in identifying prefer-ence relationships among the variables. Finally, thefeasible systems are evaluated by incorporating theprioritized constraints into the decision makingprocess. Based on this evaluation, the best system isselected. If the selected system does not meet thedecision-maker's requirements for reasons beyond

120 S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129

the scope of this process, then new systems and/orother performance variables are considered for there-evaluation process.

This paper aims at the "nal stage of the IDEF0model, which is evaluation of feasible candidates.The evaluation process is performed by utilizinga combination of CRDEA and new methodologicalextensions in DEA. The primary advantage of thismethodology is that it e!ectively integrates thedecision-maker's preferences on output measures.The rest of the paper is organized as follows. Sec-tion 2 provides a literature review of the modelsused in AMT evaluation. Section 3 provides anintroduction to the CRDEA model and its exten-sions. Section 4 presents the performance evalu-ation of industrial robots. Finally, the conclusionsand future study are presented in Section 5.

2. Literature review

Several models have been proposed in the litera-ture on evaluation, justi"cation and selection ofAMT. Le#ey [2], Leung and Tanchoco [3], Proc-tor and Canada [4], and Sarkis [5] have presentedreviews on the evolution of justi"cationmethodologies of capital projects in general, andmanufacturing technology in particular. Proposedmethodologies have included case studies, empiri-cal research, analytical, and simulation modeling.In this paper, we focus on the case of AMTs.

Canada [6] developed a weighted model forevaluating computer integrated manufacturing(CIM) systems by incorporating "nancial measuresand certain intangible factors. Huang and Ghand-foroush [7] evaluated industrial robot vendorsbased on economic measures and some subjectivefactors. They assigned speci"c weights to objectiveand subjective factor measures in identifying theoptimal choice. However, their model did not in-clude any technical robot performance measures.Imany and Schlesinger [8] compared linear goalprogramming (LGP) and ordinary least-squares(OLS) methods for robot selection. The perfor-mance measures included by them are cost, loadcapacity, velocity, and repeatability of the robot.They concluded that LGP model provides morestable results than OLS approach in the presence of

outliers. In addition, they stated that when severalcon#icting goals need to be considered, weightedpriorities should be assigned to robot parameters.

Frazelle [9] used the analytical hierarchy process(AHP) for evaluating material handling alterna-tives. According to Frazelle, a major advantage ofAHP over simple weighted evaluation techniques isits accommodation of decision maker's inconsisten-cies on preference judgement. Siha [10] utilizedmulti-goal programming (0}1) and AHP for robotselection. The model included four priority levelswith various goals in each level. Equal importancewas assigned to robot performance measures suchas repeatability, reach, and cost. Maimon andFisher [11] developed an integer programmingmodel and a rule based expert system for robotselection. Seidmann et al. [12] used a combinationof multi-goal programming and AHP for the sameproblem.

Several researchers have utilized multi-attributedecision making (MADM) models for technologyselection [13}16]. Most MADM models in AMTevaluation assume that preferences for technologyattributes are independent of each other. But this isnot true in case of AMTs such as robots wherecertain performance measures get worse as invest-ment cost decreases. Hence, the mutual preferentialindependence assumption of these models may beviolated. Khouja [17] used DEA and MADMmodel together for technology selection. Althoughthis two-phase model is innovative, there are cer-tain issues associated with the traditional DEAmodel used in this approach. The primary limita-tion of this model in the context of technologyselection is that the simple radial e$ciency scoresobtained from it may not lead to the selection of thebest choice. Shang and Sueyoshi [18] utilizeda combination of AHP and DEA for selection of#exible manufacturing systems. Their model usedAHP to restrict factor weights in DEA. However,they could not solve the di$culty with multipleoptimum solutions.

Because of the complexity of the decision makingprocess involved in AMT selection, all theaforementioned literature relied on some form ofprocedure that assigns weights to various perfor-mance measures. The primary problem associatedwith arbitrary weights is that they are subjective,

S. Talluri, K.P. Yoon / Int. J. Production Economics 66 (2000) 119}129 121

and it is often a di$cult task for the decisionmaker to accurately assign numbers to preferences.Narasimhan and Vickery [19] pointed that it is adaunting task for the decision-maker to assess weight-ing information as the number of performance criteriaincreased. Therefore, a more robust mathematicaltechnique that does not demand too much and tooprecise information, i.e., ordinal preferences insteadof cardinal weights, from the decision-maker canstrengthen the AMT evaluation process.

This paper depicts the AMT selection processthrough an IDEF0 functional model. It proposesa combination of a CRDEA model and a newmethodological extension in DEA, while allowingfor the incorporation of decision maker's prefer-ences. The chief advantage of the CRDEA model isthat it does not demand exact weights from thedecision-maker. It simply requires priority orderamong performance criteria.

3. Cone-ratio DEA model and extensions

DEA is a mathematical programming technique,proposed by Charnes et al. [20], which evaluatesthe relative e$ciencies of a homogeneous set ofdecision making units (DMUs) in the presence ofmultiple inputs and outputs. Over the years severalextensions and new models have been proposed toimprove the discriminatory power of DEA.CRDEA is one such model that allows for integra-tion of decision maker's preferences.

3.1. Traditional CRDEA model

Thompson et al. [21] introduced the concept ofassurance regions (ARs) in DEA. According tothem, ARs can be classi"ed into cone-ratio ARsand non-cone-ratio ARs. Polyhedral cone-ratiosare usually expressed in intersection form or in sumform [22]. Non-cone-ratio ARs can be expressed aslinked-cones, which link inputs and outputs, andalso as non-Archimedean and lower bound multi-pliers. A parallel study conducted by Charnes et al.[22] lead to the development of the CRDEAmodel. In the CRDEA model, ARs are speci"ed byde"ning bounds on weights that re#ect the relativeimportance of di!erent inputs and outputs. These

weight constraints are represented in the form oflinear inequalities and appended to the ratio DEAmodel. The following formulation is the CRDEAmodel with the input and output cones speci"ed by(1.3) and (1.4), respectively:

max+s

k/1vkykp

+mj/1

ujxjp

(1.1)

s.t.

+sk/1

vkyki

+mj/1

ujxji

)1 ∀i, (1.2)

aju1)u

j)b

ju1, j"2, 3,2, m, (1.3)

akv1)v

k)b

kv1, k"2, 3,2, s, (1.4)

vk, u

j*0 ∀k, j, (1.5)

i"1,2, n, j"1,2, m, k"1,2, s,

where p is the test DMU which is being evaluated,s represents the number of outputs, m representsthe number of inputs, n is the number of decisionmaking units, y

kiis the amount of output k prod-

uced by DMU i, xji

is the amount of input j used byDMU i, v

kis the weight given to output k, u

jis the

weight given to input j, and a, b, a, b are non-nega-tive scalars.

The above fractional program can be convertedto the linear programming problem shown as (2):

maxs+

k/1

vkykp

,

s.t.

m+j/1

ujxjp"1, (2)

s+k/1

vkyki!

m+j/1

ujxji)0 ∀i,

aju1!u

j)0, j"2, 3,2, m,

bju1!u

j*0, j"2,3,2, m,

akv1!v

k)0, k"2, 3,2, s,

bkv1!v

k*0, k"2, 3,2, s,

vk, u

j*0 ∀k, j.

This LP model is solved n times in estimating therelative e$ciencies of all the DMUs. The modelallows each DMU to e!ectively select optimalweights, within the speci"ed weight restrictions,


that maximize the output to input ratio, but at thesame time the constraint set prevents the e$cien-cies of the n DMUs computed with these weightsfrom exceeding a value of ònea. A relative e$cien-cy score of ònea indicates that the DMU underconsideration is e$cient, whereas a score less thanònea indicates that it is ine$cient.

The above CRDEA model has certain advant-ages over the ratio DEA model. The e$ciency scoreobtained from the ratio DEA model is referred to asthe simple radial e$ciency. In the estimation ofsimple radial e$ciency, the ratio model allows forunrestricted factor weights (v

kand u

j). This may

result in allowing a DMU to achieve a high relativee$ciency score by involving in unreasonableweighting scheme [23,24]. Such DMUs heavilyweigh few favorable measures and completelyignore the other inputs and outputs. To overcomethis problem CRDEA model allows for weightrestrictions, which signi"cantly improve the dis-criminatory power of DEA.

Although the cone restrictions e!ectively dis-criminate between e$cient and ine$cient DMUs,they do not allow for a ranking of e$cient units.This poses a problem to the decision-maker insituations where there are multiple e$cient units.Cross-e$ciency analysis in DEA can be utilized insuch situations to discriminate between e$cientunits [25]. The cross e$ciency matrix (CEM) pro-vides information on how well a DMU is perform-ing with respect to the optimal DEA weights ofother DMUs. The element in the ith row and jthcolumn of the matrix indicates how well DMU j isperforming when evaluated with the optimalweights of DMU i. A DMU with several highe$ciencies along its column in the CEM is con-sidered to be a superior performer, and a DMUwith several low e$ciencies can be regarded asa relatively poor performer. The column means canbe computed as an index to identify the best choice.

3.2. Methodological extensions

Although the CRDEA model e!ectively inte-grates decision maker's preferences into the evalu-ation process, it does not solve the problem ofmultiple optimum solutions in DEA. This problemoccurs in DEA if there are more than one set of

input/output weights that maximize the e$ciencyof a DMU. In such situations the usefulness ofCEM, which is derived from these weights, will beundermined. To overcome these problems and toprovide a robust set of weights that not only maxi-mize the e$ciency of a DMU but also minimize thee$ciencies of its competitors we provide method-ological extensions to the CRDEA model case. Theformulation shown as problem (3) is an extensionto the Doyle and Green's [26] aggressive formula-tion:

∀c

mins+k/1Avck +

ecEp

c

yke

cB (3.1)

s.t.

m+j/1Aucj +

ecEp

c

xje

cB"1, (3.2)

s+k/1

vckyke

c!

m+j/1

ucjxje

c)0 ∀ecOpc, (3.3)

s+k/1

vckykp

c!

m+j/1

ucjxjp

c"0, (3.4)

acjuc1!uc

j)0, j"2, 3,2, m, (3.5)

bcjuc1!uc

j*0, j"2, 3,2, m, (3.6)

ackvc1!vc

k)0, k"2, 3,2, s, (3.7)

bckvc1!vc

k*0, k"2, 3,2, s, (3.8)

vck, uc

j*0 ∀k, j, (3.9)

where c"1,2, r, and r represents the number ofcone scenarios, ec is an index that represents thenumber of e$cient DMUs with respect to cone c,and pc is the test DMU. The other notation forinputs and outputs is same as before.

Expression (3.1) represents the output of thecomposite DMU constructed from all the CRDEAe$cient DMUs, with respect to cone c, except testDMU pc. Expression (3.2) represents the input ofthe composite DMU, and (3.3) represents thenormalization constraints. The constraint thatblankets the e$ciency score of DMU pc, which isobtained from problem (2), is shown as (3.4).Constraints (3.5) to (3.8) are the cone restrictionsspeci"ed by the decision-maker.


Table 1List of robots and their attributes [6]

Robot Cost($10000)

Repeat(mm)

Load(kg)

Velocity(m/s)

1 6.00 0.20 6 1.502 9.00 0.40 60 2.903 9.50 0.08 28 1.004 12.50 1.00 60 1.205 7.00 0.05 20 2.506 7.20 1.25 115 0.907 4.25 0.10 50 0.708 5.80 0.15 15 1.509 11.00 0.025 10 0.67

10 5.00 0.20 40 1.0011 4.00 0.50 10 0.5012 8.50 1.25 100 0.9013 9.00 0.25 70 0.65

Table 2Scaled robot data set

Robot Cost Repeat Load Velocity

1 0.789 0.590 0.134 1.2302 1.184 0.295 1.336 2.3773 1.250 1.474 0.623 0.8204 1.645 0.118 1.336 0.9845 0.921 2.358 0.445 2.0496 0.947 0.094 2.560 0.7387 0.559 1.179 1.113 0.5748 0.763 0.786 0.334 1.2309 1.447 4.717 0.223 0.549

10 0.658 0.590 0.890 0.82011 0.526 0.236 0.223 0.41012 1.118 0.094 2.226 0.73813 1.184 0.472 1.558 0.533

Problem (3) identi"es weights that not onlymaximize the e$ciency score of a DMU, but alsominimize the e$ciencies of all other CRDEA e$-cient DMUs in some sense (by minimizing thee$ciency of the composite DMU). We did notinclude the ine$cient DMUs in problem (3), be-cause if a system is ine$cient then it is not con-sidered as a potential candidate for selection.

The primary contribution of the above model isthat it provides robust input/output weights foreach of the e$cient DMUs that can be utilized forcross-e$ciency calculations to discriminate and se-lect DMUs with best operating practices.

4. Performance evaluation of industrial robots

The robot data set published in Siha [10], shownin Table 1, is utilized to illustrate the use of theCRDEA model for technology selection. Thirteenrobots are considered as the feasible alternatives.The robot performance measures considered forthe analysis are cost, repeatability, load capacity,and velocity. These measures are not exhaustive byany means, but frequently used in robot's perfor-mance evaluation. Repeatability can be de"ned asthe ability of the robot to return repeatedly toa learned point in its work envelope, which isexpressed in millimeters (mm). Load capacity inkilograms (kg) is the maximum load that the robot

can lift, which includes the actual load and thegripper weight. Velocity of the robot is the distancecovered by the robot-arm per unit time (m/s).

In this analysis, cost is treated as an input to theCRDEA model, because from the decision maker'sstandpoint it represents the investment required topurchase the system. Repeatability, load capacity,and velocity are treated as outputs, because theyare measures that indicate the ability of the systemin performing various tasks.

In DEA, large value of an output is considered tobe better than small. Since smaller values of re-peatability indicate better performance than largervalues, we have used inverse of repeatability in ourcomputations. Also, we have re-scaled the entiredata set by dividing each performance variablevalue by its respective mean to remove any scalee!ects, especially when weight restrictions are spe-ci"ed in the CRDEA model. This is a normalizationprocedure that is commonly used in DEA compu-tations. The scaled data set is shown in Table 2.

The models discussed and proposed in this paperare applied in a sequential manner. Initially, thelimitations of the ratio DEA model (without weightrestrictions) are demonstrated. The advantages ofthe CRDEA model over the ratio DEA model arethen discussed. Finally, the results of the CRDEAmodel are utilized in problem (3) in deriving robustinput/output weights, which are used for the "nalselection process.


Table 4Simple and cone-ratio e$ciencies of robots

Robot EFF Cone 1 Cone 2 Cone 3 Cone 4 Cone 5 Cone 6 Meanv1*v

2*v

3v1*v

3*v

2v2*v

1*v

3v2*v

3*v

1v3*v

1*v

2v3*v

2*v

1

1 0.700 0.469 0.481 0.469 0.542 0.700 0.700 0.5602 1.000 0.642 0.642 0.657 0.911 0.902 1.000 0.7923 0.472 0.472 0.452 0.451 0.448 0.443 0.443 0.4524 0.427 0.282 0.281 0.344 0.407 0.281 0.427 0.3375 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.0006 1.000 0.697 0.679 1.000 1.000 0.679 1.000 0.8437 1.000 1.000 0.973 1.000 1.000 0.973 1.000 0.9918 0.737 0.585 0.585 0.588 0.653 0.724 0.737 0.6459 1.000 1.000 1.000 0.833 0.720 0.760 0.720 0.839

10 0.829 0.669 0.663 0.682 0.801 0.663 0.829 0.71811 0.400 0.313 0.313 0.318 0.373 0.350 0.400 0.34512 0.767 0.531 0.519 0.747 0.762 0.519 0.767 0.64113 0.532 0.422 0.415 0.532 0.532 0.411 0.530 0.474

Table 3DEA LP formulation for robot selection

Max 0.59v1#0.134v

2#1.23v

3s.t 0.789u

1"1

0.59v1#0.134v

2#1.23v

3!0.789u

1)0

0.295v1#1.336v

2#2.377v

3!1.184u

1)0

1.474v1#0.623v

2#0.82v

3!1.25u

1)0

0.118v1#1.336v

2#0.984v

3!1.645u

1)0

2.358v1#0.445v

2#2.049v

3!0.921u

1)0

0.094v1#2.56v

2#0.738v

3!0.947u

1)0

1.179v1#1.113v

2#0.574v

3!0.559u

1)0

0.786v1#0.334v

2#1.23v

3!0.763u

1)0

4.717v1#0.223v

2#0.549v

3!1.447u

1)0

0.59v1#0.89v

2#0.82v

3!0.658u

1)0

0.236v1#0.223v

2#0.41v

3!0.526u

1)0

0.094v1#2.226v

2#0.738v

3!1.118u

1)0

0.472v1#1.558v

2#0.533v

3!1.184u

1)0

v1, v

2, u

1, u

2*0

4.1. Ratio DEA model results

Initially, the ratio DEA model without theweight restrictions is used and simple radial e$-ciency scores are obtained for all the robots. TheDEA formulation with robot 1 as the test DMU isshown in Table 3. The e$ciency scores of the thir-teen robots are shown under the heading EFF inTable 4. DEA identi"ed robots 2, 5, 6, 7 and 9 to bee$cient with a relative e$ciency score of `onea.The other eight robots were ine$cient with e$cien-cy scores of less than `onea. These evaluations did

not include any decision-maker's preferences(weight restrictions).

It is a di$cult task for the decision-maker toselect the optimal choice from the "ve simple radiale$cient robots because:

(i) There may be robots indulging in an inappro-priate weight scheme among the "ve e$cientrobots. Thus, an arbitrarily selected e$cientrobot may be a sub-optimal choice. Also, theselected robot may not be the one that bestmeets the decision-maker's requirements.

(ii) Alternately, the decision-maker may have toassign speci"c weights to the attributes of the"ve e$cient robots in selecting the optimalchoice. But from earlier discussions, it may bea di$cult task to obtain exact weights for eachof the attributes.

Thus, using simple radial e$ciency scores alonemay not guarantee the selection of the best robot.Hence, we need to modify the model to allow forthe decision-maker's preferences.

The competitive priorities considered in this pa-per are quality, lead-time, and #exibility. Qualitycan be e!ectively linked to the repeatability of therobot, because repeatability is a measure of howwell the robot performs repetitive operations. Higherrepeatability results in better consistency, whichresults in higher quality of products. Similarly,lead-time is related to the velocity of the robot.


Table 5Cones utilized in the decision making process

Conenumber

Prioritization ofperformancevariables

Preferencerelationships

CRDEAconstraints

1 Repeatability,Load capacity,Velocity

<1*<

2*<

3<

1!<

2*0

<2!<

3*0

2 Repeatability,Velocity,Load capacity

<1*<

3*<

2<

1!<

3*0

<3!<

2*0

3 Load capacity,Repeatability,Velocity

<2*<

1*<

3<

2!<

1*0

<1!<

3*0

4 Load capacity,Velocity,Repeatability

<2*<

3*<

1<

2!<

3*0

<3!<

1*0

5 Velocity,Repeatability,Load capacity

<3*<

1*<

2<

3!<

1*0

<1!<

2*0

6 Velocity,Load capacity,Repeatability

<3*<

2*<

1<

3!<

2*0

<2!<

1*0

Table 6CEM for Cone 1 analysis

Robot 5 7 9

5 1.000 0.973 0.7207 0.742 1.000 0.8339 0.785 0.647 1.000

Mean 0.843 0.873 0.851

Higher velocity of robot-arm results in lower pro-cessing times, which decreases the overall productlead-time. Finally, higher load capacity results inhigher load #exibility. This is because the ranges ofload capacities that can be handled by the robot arehigher thereby resulting in improved #exibility.Based on this reasoning, we have selected six pos-sible cone scenarios (3! cases), shown in Table 5,where each of them can be represented as a con-straint set in the CRDEA model. For example,cone 1 represents the situation where the decisionmaker sets quality, #exibility, and lead time in thatorder as the competitive priorities thereby allowinghigher weight for repeatability (<

1) over load capa-

city (<2), and load capacity (<

2) over velocity (<

3).

This prioritization is represented as a set of prefer-ence relationships, which are incorporated into theCRDEA model as linear constraints. Although inthe construction of the cones we did not includeactual managerial information in terms of more

speci"c and appropriate bounds, this information ifavailable can be easily integrated into the CRDEAmodel. However, we did specify cones that aree!ectively linked to the competitive priorities, andallow for increased weights on higher priority vari-ables.

4.2. CRDEA model results

Cone 1 analysis depicts the scenario where thedecision-maker sets the competitive prioritiesthereby allowing higher weight for repeatability(<

1) over load capacity (<

2), and load capacity

(<2) over velocity (<

3). The results are depicted in

Table 4 under the heading `Cone 1a. Robots 5, 7,and 9 are identi"ed to be e$cient with a relativee$ciency score of `onea. All the other robots areine$cient with relative e$ciency scores of less than`onea. Thus, the decision-maker must identify thebest robot from the three e$cient choices.

Problem (3) is utilized to identify input/outputweights for Robots 5, 7, and 9, which are used in theevaluation of the CEM. The corresponding CEMof the order 3]3 is depicted in Table 6. Based onthis CEM, the column means of the CEM forRobots 5, 7, and 9 are 0.843, 0.873, and 0.851,respectively. Robot 7 achieved the highest meancross-e$ciency score of 0.873. Thus, the optimalchoice for the decision-maker is Robot 7, which isevaluated to be the best performer by the other twoe$cient robots.

Before selecting Robot 7 the decision-makerneeds to make sure that it closely satis"es otherfactors such as worker approval, #oor space re-quirements, vendor responsiveness, etc. If the se-lected robot violates any of these factorssigni"cantly then the next best option must be


Fig. 2. Comparison of robot e$ciencies.

considered. The decision-maker can identify thenext best option based on the highest cross-e$cien-cy mean, which in this case is Robot 9.

In cone 2 analysis the decision-maker allowshigher weight for repeatability (<

1) over velocity

(<3), and velocity (<

3) over load capacity (<

2).

CRDEA identi"ed Robots 5 and 9 to be e$cient,and the other eleven robots are evaluated to beine$cient. The cross-e$ciency means of the Robots5 and 9 are 0.893 and 0.860, respectively. Thus, thedecision-maker must select Robot 5 as the optimalchoice.

In cone 3 analysis, we assume that the decision-maker sets the competitive priorities thereby allow-ing higher weight for load capacity (<

2) over

repeatability (<1), and repeatability (<

1) over

velocity (<3). CRDEA identi"ed Robots 5, 6 and

7 to be e$cient, and the other ten robots are evalu-ated to be ine$cient. The cross-e$ciency means ofthe Robots 5, 6, and 7 are 0.640, 0.788, and 0.903,respectively. Therefore the optimal choice for thedecision-maker is Robot 7.

Cone 4 analysis allows for higher weight for loadcapacity (<

2) over velocity (<

3), and velocity (<

3)

over repeatability (<1). CRDEA identi"ed Robots

5, 6 and 7 to be e$cient, and the other ten robotsare evaluated to be ine$cient. The cross-e$ciencymeans of the Robots 5, 6, and 7 are 0.616, 0.893, and0.903, respectively. Thus, the decision-maker mustselect Robot 7 as the optimal choice.

Cone 5 analysis illustrates the situation wherethe decision-maker allows higher weight for velo-city (<

3) over repeatability (<

1), and repeatability

(<1) over load capacity (<

2). In this analysis only

Robot 5 is identi"ed to be e$cient. Since no otherrobot is e$cient, there is no need for cross-e$cien-cy analysis. The decision-maker can select Robot5 as the optimal choice.

Finally, in cone 6 analysis the decision-makerallows higher weight for velocity (<

3) over load

capacity (<2), and load capacity (<

2) over repeata-

bility (<1). CRDEA identi"ed Robots 2, 5, 6, and

7 to be e$cient. The mean cross-e$ciency means ofRobots 2, 5, 6, and 7 are 0.886, 0.944, 0.713, and0.760, respectively. So, the decision-maker can se-lect Robot 5 as the optimal choice.

In summary, the selected choice is dependent onthe competitive priorities set. This analysis allows


the decision-maker to identify the optimal choicebased on the critical performance variables andsystem requirements that are vital for satisfying thecorporate goals and long-term strategy of the or-ganization. If the decision maker is interested ina good overall performer then they have to go withRobot 5, which performed consistently well acrossall cones and achieved a mean score (mean of allsix cone e$ciencies) of 1.000, shown in Table 4under the heading `Meana. Fig. 2 helps facilitatethe comparison of the 13 robots across all the sixcone scenarios. It clearly demonstrates the variabil-ity and consistency in the performance of the ro-bots with respect to the decision-maker'spreferences.

5. Conclusions and future study

In this paper a CRDEA model is proposed forAMT evaluation and selection. The strength of themodel is in incorporating the decision-maker's in-put and e!ectively discriminating between e$cientand ine$cient technologies. The limitations of theratio DEA model are illustrated, and its results arecompared to that of the CRDEA model. The paperalso presents a new methodological extension inDEA to improve the discriminatory power of theCRDEA model.

As an extension to this study, it will be interest-ing to investigate the impact of the linked-conestructures, linking inputs and outputs, on the deci-sion making process. However, the decision-makermust be cautious when using these linked-conestructures because of possible infeasibility prob-lems. For more information on application of lin-ked-cones, see Thompson et al. [27].

One of the limitations of this study is that thecone restrictions are not derived from actual mana-gerial experiences, but we did provide cones thatare e!ectively linked to the competitive priorities.Although it may be a di$cult task to obtain coneratios from actual managerial practice, it is relative-ly easier to obtain these types of bounds thanspecifying exact weights. We would like to pointout that this model can easily integrate suchbounds into the decision making process, if avail-able. Another limitation of this research is that the

model does not allow for the incorporation ofqualitative factors into the selection process.

Although DEA was applied extensively in evalu-ating the e$ciencies of hospitals, schools, banks,health insurance, and other pro"t and non-pro"torganizations, its application in production andoperations "eld has been very limited. This paperillustrates its importance as a decision making toolin the area of technology management.

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A cone-ratio DEA approach for AMT justification

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