# Supplier Selection by the Pair of Nondiscretionary Factors-Imprecise Data Envelopment Analysis Models

**ABSTRACT** Discretionary models for evaluating the efficiency of suppliers assume that all criteria are discretionary, that is, controlled by the management of each supplier and varied at its discretion. These models do not assume supplier selection in the conditions that some factors are nondiscretionary. The objective of this paper is to propose a new pair of nondiscretionary factors-imprecise data envelopment analysis (NF-IDEA) models for selecting the best suppliers in the presence of nondiscretionary factors and imprecise data. A numerical example demonstrates the application of the proposed method. Yes Yes

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**ABSTRACT:**Supplier selection plays a key role in an organization because the cost of raw material constitutes the main cost of the final product. Selecting an appropriate supplier is now one of the most important decisions of the purchasing department. This decision generally depends on a number of different criteria. The objective of this paper is to propose a data envelopment analysis methodology that considers both undesirable outputs and imprecise data simultaneously. The proposed model is applied in supplier selection problem. A numerical example demonstrates the application of the proposed method. Yes YesInternational Journal of Advanced Manufacturing Technology 01/2010; · 1.78 Impact Factor - SourceAvailable from: SSRN[Show abstract] [Hide abstract]

**ABSTRACT:**This paper depicts the media selection problem through a Data envelopment analysis (DEA) model, while allowing for the incorporation of both flexible factors and imprecise data. A numerical example demonstrates the application of the proposed method.Journal of the Operational Research Society 09/2011; · 0.99 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Reaction pathway information of transition states and intermediate species is crucial for understanding the adsorption mechanism of pollutants at mineral-water interfaces. However, it has been difficult to obtain such information using existing experiments. Here, the activation barriers, transition states, intermediate species and surface complexes of arsenate adsorption on TiO(2) surfaces were studied using DFT-based reaction pathway calculations. The results indicated that the bidentate binuclear (BB) adsorption structure was formed through a monodentate mononuclear (MM) metastable-equilibrium adsorption (MEA) state. A two-step adsorption mechanism was proposed on the basis of the detailed picture of bond breaking and bond formation during each reaction step. When the adjacent surface sites were occupied, the transform from MM mode to BB mode was greatly inhibited so that both MM and BB coexisted in the equilibrium adsorption sample. The BB complex was energetically more stable than the MM complex, and so, the adsorption irreversibility was fundamentally related to the ratio BB:MM in the final equilibrium state. This mechanism may also explain the initial concentration effect, where, for the given adsorption experiment of arsenate on TiO(2) under the same thermodynamic conditions, both equilibrium constants and the BB:MM ratio in equilibrium adsorption samples changed with the reaction kinetics.Journal of Colloid and Interface Science 08/2011; 364(2):476-81. · 3.55 Impact Factor

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Supplier Selection by the Pair of Nondiscretionary Factors-Imprecise

Data Envelopment Analysis Models

Reza Farzipoor Saen

Department of Industrial Management, Faculty of Management and Accounting, Islamic Azad University - Karaj Branch,

Karaj, Iran, P. O. Box: 31485-313

Tel: 0098 (261) 4418144-6

Fax: 0098 (261) 4418156

E-mail: farzipour@yahoo.com

Abstract

Discretionary models for evaluating the efficiency of suppliers assume that all criteria are

discretionary, i.e., controlled by the management of each supplier and varied at its discretion. These

models do not assume supplier selection in the conditions that some factors are nondiscretionary. The

objective of this paper is to propose a new pair of Nondiscretionary Factors-Imprecise Data

Envelopment Analysis (NF-IDEA) models for selecting the best suppliers in the presence of

nondiscretionary factors and imprecise data. A numerical example demonstrates the application of the

proposed method.

Keywords: Nondiscretionary factors, Imprecise data envelopment analysis, Supplier selection

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Introduction

Supplier selection models are based on cardinal data with less emphasis on ordinal data. However,

with the widespread use of manufacturing philosophies such as Just-In-Time (JIT), emphasis has

shifted to the simultaneous consideration of cardinal and ordinal data in supplier selection process. On

the other hand, discretionary models for evaluating the efficiency of suppliers assume that all criteria

are discretionary, i.e., controlled by the management of each supplier and varied at its discretion. Thus,

failure of a supplier to produce maximal output levels with minimal input consumption results in a

decreased efficiency score. In any realistic situation, however, there may exist exogenously fixed or

nondiscretionary criteria that are beyond the control of a management. In an analysis of a network of

fast food restaurants, Banker and Morey (1986) illustrate the impact of exogenously determined inputs

that are not controllable. In their study, each of the 60 restaurants in the fast food chain consumes six

inputs to produce three outputs. The three outputs (all controllable) correspond to breakfast, lunch, and

dinner sales. Only two of the six inputs, expenditures for supplies and expenditures for labor, are

discretionary. The other four inputs (age of store, advertising level, urban/rural location, and

presence/absence of drive-in capability) are beyond the control of the individual restaurant manager.

Their analysis clearly demonstrates the value of accounting for the nondiscretionary character of these

inputs explicitly in the Data Envelopment Analysis (DEA) models they employ; the result is

identification of a considerably enhanced opportunity for targeted savings in the controllable inputs and

targeted increases in the outputs. In the case of supplier selection, distance and supply variety are

generally considered nondiscretionary criterion.

Weber (1996) applied DEA in supplier evaluation for an individual product and demonstrated the

advantages of applying DEA to such a system. In Weber’s study, six vendors supplying an item to a

baby food manufacturer were evaluated. Significant reductions in costs, late deliveries, and rejected

materials can be achieved if inefficient vendors can become DEA efficient.

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When suppliers are compared for their overall performances, an aggregate evaluation relevant to

the considerations of a purchasing firm needs to be conducted. Such an overall performance evaluation

of suppliers should be based on performance measures for all part types supplied to the purchasing

company. A potential use of an overall performance evaluation of suppliers is to provide benchmarking

data for reducing the number of suppliers, which in turn results in benefits including reduction in costs

of parts and order processing, and better partnership with suppliers.

This paper depicts the supplier selection process through an Imprecise Data Envelopment Analysis

(IDEA) model, while allowing for the incorporation of nondiscretionary factors. The objective of this

paper is to propose a new pair of Nondiscretionary Factors-Imprecise Data Envelopment Analysis (NF-

IDEA) models for selecting the best suppliers in the presence of nondiscretionary factors and imprecise

data.

Literature review

Some mathematical programming approaches have been used for supplier selection in the past.

Zeng et al, (2006) considered a simplified partner selection problem which takes into account only the

bid cost, the bid completion time of subprojects, and the due date of the project. They modeled the

problem as a nonlinear integer programming problem and proved that the decision problem of the

partner selection problem is NP-complete. Then they analysed some properties of the partner selection

problem and construct a branch and bound algorithm.

Hajidimitriou and Georgiou (2002) presented a quantitative model, based on the Goal Programming

(GP) technique, which uses appropriate criteria to evaluate potential candidates and leads to the

selection of the optimal partner (supplier).

ebi

,C

and Bayraktar (2003) proposed an integrated model

for supplier selection. In their model, supplier selection problem has been structured as an integrated

Lexicographic Goal Programming (LGP) and Analytic Hierarchy Process (AHP) model including both

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quantitative and qualitative conflicting factors. Karpak et al, (2001) presented one of the "user-

friendly" multiple criteria decision support systems-Visual Interactive Goal programming (VIG). VIG

facilitates the introduction of a decision support vehicle that helps improve the supplier selection

decisions. To take into account both cardinal and ordinal data in supplier selection, Wang et al, (2004)

developed an integrated AHP and Preemptive Goal Programming (PGP) based methodology.

However, one of the GP problems arises from a specific technical requirement. After the

purchasing managers specify the goals for each selected criterion (e.g., amount of price, quality level,

etc), they must decide on a preemptive priority order of these goals, i.e., determining in which order the

goals will be attained. Frequently such a priori input might not produce an acceptable solution and the

priority structure may be altered to resolve the problem once more. In this fashion, it may be possible to

generate a solution iteratively that finally satisfies the Decision Maker (DM). Unfortunately, the

number of potential priority reorderings may be very large. A supplier selection problem with five

factors has up to 120 priority reorderings. Going through such a laborious process would be costly and

inefficient.

Sha and Che (2006) presented a multi-phased mathematical approach called the Hybrid Multi-

phased-based Genetic Algorithm (HMGA) for network design of supply chain. From the point of view

of network design, the important issues are to find suitable and quality companies, and to decide upon

an appropriate production/distribution strategy. It is based on various methodologies that embrace

Genetic Algorithms (GAs), AHP, and the Multi-Attribute Utility Theory (MAUT) to simultaneously

satisfy the preferences of suppliers and customers at each level of the supply chain network. Bayazit

(2006) provided a good insight into the use of the Analytic Network Process (ANP) that is a multiple

criteria decision making methodology in evaluating supplier selection problems. Dulmin and Mininno

(2003) presented a proposal for applying a decision model to the final vendor-rating phase of a process

of supplier selection. Their model uses a Multiple Criteria Decision Aid (MCDA) technique

(PROMETHEE 1 and 2), with a high-dimensional sensitivity analysis approach. They tried to explain

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how an outranking method and PROMETHEE/GAIA techniques, provides powerful tools to rank

alternatives and analysed the relations between criteria or between DMs. Bhutta and Huq (2002)

illustrated Total Cost of Ownership (TCO) and AHP approaches and provided a comparison. They

concluded that TCO is better suited to those situations where cost is of high priority and detailed cost

data are available to make comparisons. In the case of AHP, it is better suited to solve and decide

between suppliers when several conflicting goals exist and, though cost may be an important factor, it

is not the overriding one.

However, AHP has two main weaknesses. First subjectivity of AHP is a weakness. Second AHP

could not include interrelationship within the criteria in the model.

Chen et al, (2006) presented a fuzzy decision making approach to deal with the supplier selection

problem in supply chain system. They used linguistic values to assess the ratings and weights for the

criteria. These linguistic ratings can be expressed in trapezoidal or triangular fuzzy numbers. Then, a

hierarchy Multiple Criteria Decision Making (MCDM) model based on fuzzy sets theory is proposed to

deal with the supplier selection problems in the supply chain system. According to the concept of the

Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), a closeness coefficient is

defined to determine the ranking order of all suppliers by calculating the distances to the both Fuzzy

Positive Ideal Solution (FPIS) and Fuzzy Negative Ideal Solution (FNIS) simultaneously.

Choy et al, (2002) presented an Intelligent Supplier Management Tool (ISMT) using the Case-

Based Reasoning (CBR) and Neural Network (NN) techniques to select and benchmark suppliers.

Choy and Lee (2003) suggested an intelligent Generic Supplier Management Tool (GSMT) using the

CBR technique for outsourcing to suppliers and automating the decision making process when

selecting them. Choy et al, (2004) discussed an Intelligent Supplier Relationship Management System

(ISRMS) integrating a company’s Customer Relationship Management (CRM) system, Supplier Rating

System (SRS) and Product Coding System (PCS) by the CBR technique to select preferred suppliers

during the New Product Development (NPD) process. In order to develop a flexible data access

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framework, and to support the partner selection activity, the combination of OnLine Analytical

Processing (OLAP), and CBR was proposed by Lau et al, (2005).

Lee et al, (2003) proposed a High-Quality-Supplier Selection (HQSS) model to deal with supplier

selection problems in supply chain management. In selecting a supplier, quality management factors

are considered first, and then price, delivery, etc. Linn et al, (2006) proposed a new approach to

supplier selection using a Capability index and Price Comparison (CPC) chart. The CPC chart

integrates the process capability and price information of multiple suppliers and presents them in a

single chart for the management to make supplier selection decisions.

All the aforementioned literature relied on some form of procedures that assigns weights to various

performance measures. The primary problem associated with arbitrary weights is that they are

subjective, and it is often a difficult task for the DM to accurately assign numbers to preferences. It is a

daunting task for the DM 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 DM can strengthen

the supplier evaluation process. To this end, Weber (1996) demonstrated how DEA can be used to

evaluate vendors on multiple criteria and identified benchmark values which can then be used for this

purpose. Weber et al, (2000) presented an approach for evaluating the number of vendors to employ in

a procurement situation using Multi-Objective Programming (MOP) and DEA. The approach advocates

developing vendor-order quantity solutions (referred to as supervendors) using MOP and then

evaluating the efficiency of these supervendors on multiple criteria using DEA. Recently, to select the

best suppliers in the presence of both cardinal and ordinal data, Farzipoor Saen (2007) proposed an

innovative method, which is based on IDEA. However, he did not consider the weights restrictions.

However, all the aforementioned references are based on complete discretion of decision making

criteria (factors) and do not consider supplier selection in the presence of both imprecise data and

nondiscretionary factors.

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To the best of author’s knowledge, there is not any reference that deals with supplier selection in the

conditions that nondiscretionary factor and imprecise data are present.

Proposed model for supplier selection

DEA formulations proposed by Charnes et al, (1978) (CCR model) and developed by Banker et al,

(1984) (BCC model) is an approach for evaluating the efficiencies of Decision Making Units (DMUs).

This evaluation is generally assumed to be based on a set of cardinal (quantitative) output and input

factors. In many real world applications (especially supplier selection problems), however, it is

essential to take into account the existence of ordinal (qualitative) factors when rendering a decision on

the performance of a DMU. Very often, it is the case that for a factor such as supplier reputation, one

can, at most, provide a ranking of the DMUs from best to worst relative to this attribute. The capability

of providing a more precise, quantitative measure reflecting such a factor is generally beyond the realm

of reality. In some situations, such factors can be legitimately quantified, but very often such

quantification may be superficially forced as a modeling convenience. In situations such as that

described, the data for certain influence factors (inputs and outputs) might better be represented as rank

positions in an ordinal, rather than numerical sense. Refer again to the supplier reputation example. In

certain circumstances, the information available may permit one to provide a complete rank ordering of

the DMUs on such a factor. Therefore, the data may be imprecise.

Recently, Wang et al, (2005) developed a new pair of interval DEA models for dealing with

imprecise data such as interval data, ordinal preference information, fuzzy data and their mixture.

Compared with the IDEA model developed by Cooper et al, (1999), Cooper et al, (2001a), and Cooper

et al, (2001b), their interval DEA models are much easier to understand and more convenient to use.

Also, compared with the interval DEA models developed by Despotis and Smirlis (2002), their interval

DEA models utilise a fixed and unified production frontier as a benchmark to measure the efficiencies

of all DMUs, which makes their models more rational and more reliable. Moreover, the means by

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which they treat ordinal preference information also seems more reasonable than the way Zhu (2003)

did. However, Wang et al, (2005) did not consider nondiscretionary factors.

In this section, a new pair of NF-IDEA models is proposed that can overcome the shortcoming

mentioned above, to consider nondiscretionary factors of the suppliers (DMUs) while ordinal and

cardinal data are present. The final efficiency score for each DMU will be characterised by an interval

bounded by the best lower bound efficiency and the best upper bound efficiency of each DMU.

Suppose that there are n DMUs to be evaluated. Each DMU consumes m inputs to produce s

outputs. In particular, DMUj consumes amounts Xj ={ }

ij x

of inputs (i=1, …, m) and produces amounts

Yj={ }

rj

y

of outputs (r=1, …, s). Without loss of generality, it is assumed that all the input and output

data xij and yrj (i=1, …, m; r=1, …, s; j=1, …, n) cannot be exactly obtained due to the existence of

uncertainty. They are only known to lie within the upper and lower bounds represented by the intervals

[

ij

x ,

rj

y ,

, where

0

>

and

0

>

rj

y

.

]

UL

ijx

and []

UL

rjy

L

ij x

L

In order to deal with such an uncertain situation, the following pair of linear programming models

have been developed to generate the upper and lower bounds of interval efficiency for each DMU

(Wang et al, 2005):

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. ,

r

,

) 2 (,,, 1, 0

, 1

. .

t s

. ,

r

,

) 1 (,,, 1, 0

, 1

. .

t s

11

1

1

11

1

1

ivu

njxvyu

xv

yuMax

ivu

njxvyu

xv

yu Max

ir

m

i

L

ij i

s

r

U

rjr

m

i

U

ij i

s

r

L

rjr

L

jo

ir

m

i

L

iji

s

r

U

rjr

m

i

L

iji

s

r

U

rjr

U

jo

o

o

o

o

∀≥

=≤−

=

=

∀≥

=≤−

=

=

∑

=

∑

=

∑

=

∑

=

∑

=

∑

=

∑

=

∑

=

ε

θ

ε

θ

where jo is the DMU under evaluation (usually denoted by DMUo); ur and vi are the weights assigned to

the outputs and inputs;

U

jo

θ stands for the best possible relative efficiency achieved by DMUo when all

the DMUs are in the state of best production activity, while

L

jo

θ stands for the lower bound of the best

possible relative efficiency of DMUo. They constitute a possible best relative efficiency interval

[

jo

θ ,

. ε is the non-Archimedean infinitesimal.

]

UL

joθ

In order to judge whether a DMU is DEA efficient or not, the following definition is given.

Definition 1. A DMU, DMUo, is said to be DEA efficient if its best possible upper bound efficiency

; 1

*=

jo

θ

U

otherwise, it is said to be DEA inefficient if

. 1

*<

jo

θ

U

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Now, to demonstrate how to consider nondiscretionary factors in the model, the new pair of NF-

IDEA models is proposed. The envelopment formulation (dual problem) of Models (1) and (2)

becomes

.,, 10

,,, 1

=

0

,,, 10

free,

j

=

,,, 10

) 4 (,, , 1

. .

t s

,

.,, 10

,,, 1

=

0

,, , 10

free,

j

=

,, , 10

) 3 (,, , 1

. .

t s

,

1

1

11

1

1

11

srs

mis

n

misxx

srysy

ssMinc

srs

mis

n

misxx

srysy

ssMinc

r

i

+

j

o

ij

n

j

L

ijo

U

ijo

L

rjorj

n

j

U

rj

s

r

r

m

i

io

L

jo

r

i

+

j

o

ij

n

j

L

ijo

L

ijo

U

rjorj

n

j

U

rj

s

r

r

m

i

io

U

jo

≥

=≥

≥

==−−

==−

−−=

≥

=≥

≥

==−−

==−

−−=

−

−

=

+

=

=

+

=

−

−

−

=

+

=

=

+

=

−

∑

∑

∑∑

∑

∑

∑∑

λ

θ

λθ

λ

εεθ

λ

θ

λθ

λ

εεθ

where

o θ ,

+

r

−

ij

ss

and,,

λ

are the dual variables.

o θ is the radial input shrinkage factor (eventually

to become efficiency measure) and

{ }

λ

j

λ =

is the vector of DMU loadings, determining "best

practice" for the DMU being evaluated.

U

jo

c stands for the best possible relative efficiency achieved by

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DMUo when all the DMUs are in the state of best production activity, while

L

jo

c stands for the lower

bound of the best possible relative efficiency of DMUo. They constitute a possible best relative

]

jo

c ,

. The variable

efficiency interval [

UL

joc

+

rs is shortfall amount of output r and

−

is is excess amount of

input i. From the duality theory in linear programming, for an inefficient DMUo,

0

*>

j λ

in the optimal

dual solution implies that DMUj is a unit of the peer group. A peer group of an inefficient DMUo is

defined as the set of DMUs that reach the efficiency score of 1 using the same set of weights that result

in the efficiency score of DMUo. It is the existence of this collection of DMUs that forces the DMUo to

be inefficient.

Now, the model that considers both imprecise data and nondiscretionary factors is introduced.

Suppose that the input variables may be partitioned into subsets of discretionary (D) and

nondiscretionary (N) variables. Thus,

The pair of NF-IDEA models is then finally given by

{

2 , 1

}

Φ=∩∪==

NDND

IIIImI

,,...,

Page 12

12

., , 10

,0

,0

,, , 10

free,

j

=

,0

,0

) 6 (,,, 1

. .

t s

,

., , 10

,0

,0

,, , 10

free,

j

=

,0

,0

) 5 (,,, 1

. .

t s

,

1

1

1

1

1

1

1

1

srs

Nis

Dis

n

Nisxx

Disxx

srysy

ssMinc

srs

Nis

Dis

n

Nisxx

Disxx

srysy

ssMinc

r

i

+

i

−

j

o

ij

n

j

L

ij

U

ijo

ij

n

j

L

ijo

U

ijo

L

rjorj

n

j

U

rj

s

r

r

Di

io

L

jo

r

i

+

i

−

j

o

ij

n

j

L

ij

L

ijo

ij

n

j

L

ij o

L

ijo

U

rjorj

n

j

U

rj

s

r

r

Di

io

U

jo

=≥

∈=

∈≥

≥

∈=−−

∈=−−

==−

−−=

=≥

∈=

∈≥

≥

∈=−−

∈=−−

==−

−−=

−

−

=

−

=

+

=

=

+

∈

−

−

−

=

−

=

+

=

=

+

∈

−

∑

∑

∑

∑∑

∑

∑

∑

∑∑

λ

θ

λ

λθ

λ

εεθ

λ

θ

λ

λθ

λ

εεθ

It is to be noted that the

o θ to be minimised appears only in the constraints for which

Di∈

,

whereas the constraints for which

Ni∈

operate only indirectly (as they should) because the input

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levels

ijo

x

are not subject to managerial control. Therefore this is recognised by entering all

Nixijo

∈

,

at their fixed (observed) value. Note that the slacks

Nisi

∈

−,

are omitted from the objective function.

Hence these nondiscretionary inputs do not enter directly into the efficiency measures being optimised

in (5) and (6). They can, nevertheless, affect the efficiency evaluations by virtue of their presence in the

constraints. For models (5) and (6), it is not relevant to minimise the proportional decrease in the entire

input vector. Such minimisation should be determined only with respect to the subvector that is

composed of discretionary inputs.

In order to judge whether a DMU is DEA efficient or not, the following definition is given.

Definition 2. A DMU, DMUo, is said to be DEA efficient if its best possible upper bound efficiency

; 1

*=

jo

U

c

otherwise, it is said to be DEA inefficient if

. 1

*<

jo

U

c

Therefore, one unified approach that deals with all aspects of the imprecise data and

nondiscretionary factors in a direct manner has been introduced.

Now, the method of transforming ordinal preference information into interval data is discussed, so

that the pair of NF-IDEA models presented in this paper can still work properly even in these

situations.

Suppose some input and/or output data for DMUs are given in the form of ordinal preference

information. Usually, there may exist three types of ordinal preference information: (1) strong ordinal

preference information such as yrj>yrk or xij>xik, which can be further expressed as

rkr rj

yy

χ≥

and

,

ik iij

xx

η≥

where

1

>

r

χ

and

1

>

i η

are the parameters on the degree of preference intensity provided

by decision maker; (2) weak ordinal preference information such as

rqrp

yy

≥

or

;

iqip

xx ≥

(3)

indifference relationship such as yrl = yrt or xil = xit. Since a DEA model has the property of unit-

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invariance, the use of scale transformation to ordinal preference information does not change the

original ordinal relationships and has no effect on the efficiencies of DMUs. Therefore, it is possible to

conduct a scale transformation to every ordinal input and output index so that its best ordinal datum is

less than or equal to unity and then give an interval estimate for each ordinal datum.

Now, consider the transformation of ordinal preference information about the output yrj (j=1,…, n)

for example. The ordinal preference information about input and other output data can be converted in

the same way.

For weak ordinal preference information

,

21

rnrr

yyy

≥≥≥

we have the following ordinal

relationships after scale transformation:

,

ˆ

y

ˆ

y

ˆ

y

1

21

r rnrr

σ≥≥≥≥≥

where

r

σ is a small positive number reflecting the ratio of the possible minimum value {yrj| j=1,…, n}

to its possible maximum value. As well, it can be approximately estimated by the decision maker. It is

referred as the ratio parameter for convenience. The resultant permissible interval for each

rj

y ˆ is given

by

[]

.,, 1 , 1 ,

r

ˆ

ynj

rj

=∈ σ

For strong ordinal preference information

,

21

rnrr

yyy

>>>

there is the following ordinal

relationships after scale transformation:

,

ˆ

y

and ) 1

−

, , 1(

ˆ

y

ˆ

y

,

ˆ

y

1

1,1

r rnjrrrjr

nj

σχ≥=≥≥

+

where

r

χ is a preference intensity parameter satisfying

r

χ >1 provided by the decision maker and

r

σ

is the ratio parameter also provided by the decision maker. The resultant permissible interval for each

rj

y ˆ can be derived as follows:

[

σ

]

.with,, 1,,

ˆ

y

1

r

1

r

n

r

jj

n

rr rj

nj

−−−

≤=∈χσχχ

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Finally, for an indifference relationship, the permissible intervals are the same as those obtained for

weak ordinal preference information.

Through the scale transformation above and the estimation of permissible intervals, all the ordinal

preference information is converted into interval data and can thus be incorporated into the pair of NF-

IDEA models.

In the next section, a numerical example is presented.

Numerical example

The data set for this example is partially taken from Farzipoor Saen (2007) and contains

specifications on 18 suppliers (DMUs). In particular, this example is used to show how ordinal and

bounded data, as well as nondiscretionary factors, can be combined into the one unified approach

provided by NF-IDEA. The cardinal inputs considered are Total Cost of shipments (TC) and Distance

(D). D is generally considered as a nondiscretionary input variable. Supplier Reputation (SR) is

included as a qualitative input while Number of Bills received from the supplier without errors (NB)

will serve as the bounded data output. SR is an intangible factor that is not usually explicitly included

in evaluation model for supplier. This qualitative variable is measured on an ordinal scale so that, for

instance, reputation of supplier 18 is given the highest rank, and supplier 17, the lowest. Note that, the

measures selected in this paper are not exhaustive by any means, but are some general measures that

can be utilised to evaluate suppliers. In an application of this methodology, DMs must carefully

identify appropriate inputs and outputs to be used in the decision making process. The second and third

column of Table 1 depicts the inputs and output of suppliers.

Suppose the preference intensity parameter and the ratio parameter about the strong ordinal

preference information are given (or estimated) as

,01. 0and12 . 1

=

33

=ση

respectively. To show the