Research ArticleEvaluation and Selection of Manufacturing Suppliers in B2BE-Commerce Environment

Quan Zhang Zhen Guo Feiyu Man and Jiyun Ma

School of Information Engineering ShenYang University of Technology ShenYang 110870 China

Correspondence should be addressed to Quan Zhang isqzhangsuteducn

Received 1 May 2020 Revised 24 May 2020 Accepted 1 June 2020 Published 22 June 2020

Guest Editor Lei Xie

Copyright copy 2020 Quan Zhang et al )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

)e evaluation and selection of manufacturing suppliers in B2B e-commerce environment is summed up as a multiple-attributedecision-making problem In B2B E-commerce environment some performance indicators of manufacturing suppliers presentuncertainty and could not be expressed with precise numeric values Linguistic terms preference orderings or interval numbersare commonly used to express the performances of the suppliers accurately instead of crisp values when the available informationis uncertain or incomplete )is paper proposes an approach to the selection of manufacturing suppliers in B2B E-commerceenvironment where the attribute values in decision matrix are expressed with linguistic terms preference orderings and intervalnumbers Firstly the hybrid decision matrix is normalized by calculating the grey correlation coefficients of attribute values withthe ideal values of attributes Secondly a deviation maximization model is proposed to determine the attribute weights which iscombined with those derived from the entropy method )irdly the overall values of suppliers are calculated and their rankingsare obtained Finally an example is used to illustrate the proposed approach

1 Introduction

As important components of the supply chain suppliersusually play important roles in the manufacturing process[1ndash3] )e relationships between manufacturers and sup-pliers are examined by Svensson et al [2] )e evaluationand selection of suppliers are important steps in the oper-ations of manufacturers and can be modeled as multiple-attribute decision-making (MADM) problems which in-volve some qualitative attributes for example the qualityfactor and risk factor of the suppliers

Traditionally both the qualitative attributes and thequantitative attributes are adopted in modeling the evalu-ation and selection of manufacturing suppliers [4] Howeverin B2B E-commerce environment both the qualitative at-tributes and the quantitative attributes of the suppliers showmuch more ambiguity and uncertainty than before [5 6]Natural language is introduced by Zadeh [7] and can be usedto reduce the burden of expressing subjective uncertainjudgments in a decision-making process A linguistic term isone of the easy ways for evaluations in uncertain

environment Linguistic terms are usually used to assess thequalitative attributes of the suppliers [4 8] In the meantimebecause the information available is uncertain or incom-plete preference orderings are also likely to be employed toevaluate the qualitative attributes in addition to linguisticterms Preference orderings are also the easy ways forevaluating the suppliers by describing their relative posi-tions ie the ranking orders of the suppliers against somequalitative attributes Preference orderings are used toevaluate the attribute weights in [9]

Furthermore in the global competitive situationsagainst the quantitative attributes the performance ofsuppliers would be a range of possible values with theminimum and the maximum of the utility scores [9] Inother words interval numbers are the ways of evaluatingsuppliers against some quantitative attributes [9]

It can be seen that in B2B E-commerce environmentlinguistic terms preference orderings and interval numbersare the appropriate expressions of the performance indi-cators (ie attribute values) of manufacturing suppliers Inthis case we are facing challenges when evaluating and

HindawiComplexityVolume 2020 Article ID 8690402 8 pageshttpsdoiorg10115520208690402

selecting the suppliers for the manufacturers in B2BE-commerce environment It is desirable to propose anapproach to the evaluation and selection of manufacturingsuppliers in B2B E-commerce environment when theirperformance indicators are expressed by means of suchhybrid information

However the research on evaluating the manufacturingsuppliers in B2B E-commerce environment is not socommon when their performances or attribute values aremultiple types of information such as linguistic termspreference orderings and interval numbers )e purpose ofthis paper is to develop an approach for evaluating andselecting the manufacturing suppliers in B2B E-commerceenvironment where their attribute values are expressed withlinguistic terms preference orderings and interval numbersNormalizations on the hybrid attribute values are con-ducted based on which the attribute weights are determinedso that their overall performances (overall values) are ob-tained for rankings and selections

2 Current Research and Research Objectives

)e approaches of evaluating and selecting suppliers can beclassified into three categories including (i) multiple attri-bute decision-making approaches [9ndash12] (ii) mathematicalprogramming approaches [13ndash15] and (iii) intelligent ap-proaches (eg ANN and Grey system theory)

According to the multiple-attribute decision-makingapproaches linear weighting methods and TOPSIS areusually utilized With linear weighting methods the overallvalues of every suppliers are calculated by summing up everyattribute values multiplied by their corresponding weights)e supplier(s) with the highest overall values would be se-lected TOPSIS ie the technique for order preference bysimilarity to an ideal solution was first proposed by Hwangand Yoon [16] )e TOPSIS method ranks a limited numberof alternatives according to the relative degree of proximity tothe idealized alternatives [10ndash12] )e best the alternative isthe smallest degree of proximity to the idealized alternativesGenerally when applying the linear weighting methods andTOPSIS attribute weights should be determined beforehandand the attribute weights can be assigned by decision makersor by using the AHP method subjectively

According to the mathematical programming ap-proaches usually multiple objectives are contained in theobjective function as well as some constraints are consideredsimultaneously In [13] in order to determine the bestsuppliers AHP is integrated with nonlinear and multi-objective integer programming model under quantity dis-counts and capacity and budget constraints while theobjectives of the model are maximizing the total value ofpurchase (TVP) minimizing the total cost of purchase (TCP)or maximizing TVP and minimizing TCP simultaneously In[14] a mixed-integer nonlinear program is proposed to solvedynamic supplier selection problems In [15] the problem ofsupplier selection and order allocation with multiperiodmultiproduct multisupplier andmultiobjective is generalizedas a mixed integer linear programming model where theobjectives are total inventory cost (ie delay holding and

shortage ordering and discounted purchase costs) and theconstraints are the budget and capacity limitations for bothbuyers and suppliers )e model is solved by means of apreemptive fuzzy goal programming approach

According to the intelligent approaches artificial neuralnetwork (ANN) models [17 18] and Grey theory [19ndash21] areusually employed In [17] an artificial neural network-basedpredictive model is developed for forecasting the supplierrsquosbid prices in the supplier selection negotiation process byallowing a demander to foresee the relationship between itsalternative bids and corresponding supplierrsquos next bid pricesin advance which decreases the meaningless negotiationtimes reduces the procurement cost improves the negotia-tion efficiency or shortens the supplier selection lead-time In[18] an adaptive neuro-fuzzy inference system is developedby determining the criteria and applied for supplier selections

In the recent years grey theory is applied to deal withuncertainty inherent in evaluating the suppliers while thelinguistic terms are adopted to express their attribute values[19ndash21] Although the proposed approach in [21] does notrequire any probability distribution or fuzzy membershipfunction preference orderings and interval numbers are notconsidered for the attribute values of suppliers

In [9] the TOPSISmethod is extended to the situation byallowing the input attribute values being interval numberswith the minimum and the maximum of the utility scores (arange of possible values for quantitative attributes and a listof possible grades for qualitative attributes) )e rank ordercentroid (ROC) method is used to determine the attributeweights based on the attribute ranking orders [22 23] )eminimum and the maximum outputs of the extendedTOPSIS are obtained and their averages are adopted as theoverall index for selections

)ere is rare research on tackling the supplier selectionproblems with the attribute values being linguistic termspreference orderings and interval numbers )e researchobjective of this paper is to propose a new approach to dealwith the qualitative attribute values expressed with linguisticterms and preference orderings and the quantitative attri-bute values expressed with interval numbers when evalu-ating and selecting the manufacturing suppliers in B2BE-commerce environment

)is paper is organized as follows Section 1 introducesthe research background Section 2 reviews the currentresearch on evaluating and selecting suppliers Section 3describes the evaluation and selection of suppliers as aMADM problem In Section 4 a new approach is proposedto evaluating and selecting the manufacturing suppliers inB2B E-commerce environment while the attribute valuesare expressed in linguistic terms preference orderings andinterval numbers In Section 5 an example is used to il-lustrate the proposed approach Section 6 gives the con-clusions and discussions

3 Problem Descriptions

As stated above the evaluation and selection ofmanufacturing suppliers in B2B E-commerce environmentis modeled as a MADM problem )e following notations

2 Complexity

and assumptions are used to represent the MADM problemof evaluating and selecting manufacturing suppliers

)e alternatives (ie the suppliers) are known Let S

S1 S2 Sm1113864 1113865 denote a discrete set of m(ge2) possible al-ternatives )e attributes are known and letC C1 C2 Cn1113864 1113865 denote a set of n(ge2) attributes Inorder to distinguish the qualitative and quantitative attri-butes with different characteristics the subscript of the at-tributes is divided into three categories J1 for the subscriptset of attributes with linguistic assessment values J2 for thesubscript set of attributes with preference ordering assess-ment values and J3 for the subscript set of attributes withinterval number assessment values

Let 1113957A [1113957aij]mtimesn denote the decision matrix where 1113957aij

are the assessment values for alternative Si with respect toattribute Cj i 1 m and j 1 n In this study 1113957aij

are in the forms of linguistic terms preference orderingsand interval numbers

Let W (w1 w2 wn) denote the weight vector ofn(ge 2) attributes where wj is the weight of attribute Cjwhile 1113936

nj1 wj 1 and wj gt 0 holds for j 1 n

)e problem focused in this paper is to select the bestsupplier(s) for a manufacturer in B2B E-commerce envi-ronment while their performance indicators (ie attributevalues) are linguistic terms preference orderings and in-terval numbers

4 The Proposed Approach

)e proposed approach to the problem stated in Section 3 iscomposed of three steps normalize the attribute values indifferent formats based on the grey relational degreemethod determine the attribute weights and calculate theoverall values of the alternatives (suppliers)

41 Normalize the Attribute Values in Different FormatsSince the attribute values in decision matrix 1113957A [1113957aij]mtimesn arein the formats of linguistic terms preference orderings andinterval numbers corresponding methods are developed fortransforming them into a comparable format ie the utilityvalue

411 Calculate the Grey Correlation Coefficients of LinguisticAttribute Values

Definition 1 A linguistic term 1113957T on a real-number set isdefined as a triangular fuzzy number (denoted as (u α andβ)) if its membership function μ1113957T

(R+⟶ [0 1]) is definedas

μ1113957T(x)

x minus αu minus α

x isin [α u]

x minus βu minus β

x isin [u β]

0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

where αle ule β u is the model value and α and β stand forthe lower value and the upper value of linguistic term 1113957Trespectively

Given the hybrid decision matrix 1113957A [1113957aij]mtimesn the at-tributes Cj (j isin J1) with linguistic assessment values may beof different granularities and different linguistic evaluationsets would be employed [24] )erefore a basic linguisticevaluation set TERMSETB (TERMSETB termB

0 termB1

termBg) is used to transform the linguistic assessment values

with different granularities into the comparable form )etriangular fuzzy number cB

l (μBl αB

l βBl ) corresponding to

the linguistic term termBi is defined as follows

cBl

αB0 0

uBl

l

g minus 1 0le lleg minus 1

αBl

l minus 1g minus 1

1le lleg minus 1

βBl

l + 1g minus 1

0le lleg minus 2

βBgminus1 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)

where μBl is the model value and αB

l and βBl stand for the

lower value and the upper value of cBl

With respect to the attribute values with linguistic as-sessments 1113957aij (i 1 m j isin J1) denote 1113957aij as lingij lingij

can be transformed into the fuzzy set over the basic linguisticevaluation set TERMSETB (denoted as Fij (TERMSETB))

τ lingij⟶ Fij TERMSETB1113872 1113873 i 1 m j isin J1

(3)

where Fij(TERMSETB) is the fuzzy set over the basic lin-guistic evaluation set TERMSETB as stated as follows

Fij TERMSETB1113872 1113873 termB

l cijl1113872 1113873|l isin [0 g] l 0 g1113966 1113967

(4)

where

cijl maxy

min μlingij(y) μtermB

l(y)1113882 1113883 (5)

where μlingij(y) and μtermB

l(y) denote the membership

functions of lingij and termBl respectively i 1 m l 0

g j isin J1Furthermore the fuzzy set Fij (TERMSETB) can be

transformed into a crisp value as follows

ϕ Fij TERMSETB1113872 11138731113872 1113873

1113936g

l0 l times cijl

g 1113936g

l0 cijl

i 1 m j isin J1

(6)

)us given the linguistic assessment values 1113957aij of at-tributes Cj (j isin J1) by means of the operations in (3)ndash(6)

Complexity 3

linguistic term 1113957aij is transformed into the form of crispvalue denoted as

bij ϕ Fij TERMSETB1113872 11138731113872 1113873 i 1 m j isin J1 (7)

In the meantime the positive ideal attribute value forattribute Cj (j isin J1) is defined as follows

ideal+j max1leilem

bij1113966 1113967 j isin J1 (8)

)erefore with respect to the attributes Cj (j isin J1) withlinguistic assessment values the grey relational coefficientsbetween their crisp values bij and the corresponding positiveideal attribute value ideal+j are defined as

qij miniminj dis bij ideal

+j1113872 11138731113966 1113967 + ρ maximaxj dis1 bij ideal

+j1113872 11138731113966 1113967

dis1 bij ideal+j1113872 1113873 + ρ maximaxj dis1 bij ideal

+j1113872 11138731113966 1113967

i 1 m j isin J1 (9)

where dis1(middot) is the distance function between bij and ideal+jand is defined as follows

Dis1 bij ideal+j1113872 1113873 ideal+j minus bij i 1 m j isin J1

(10)

412 Calculate the Grey Correlation Coefficients of the At-tribute Values in the Form of Preference Orderings Withrespect to the attributes Cj (j isin J2) with the assessmentvalues in the form of preference orderings denote 1113957aij(i 1 m j isin J2) as Rij and Rij is a permutation function overthe index set 1 m Alternatively Rij represents theposition of Si in the preference ordering Rij can be trans-formed into a crisp value bij as follows

bij m minus Rij

m minus 1 i 1 m j isin J2 (11)

In addition regarding the attributes Cj (j isin J2) with theassessment values in the form of preference orderings thepositive ideal attribute value for attribute Cj is defined asfollows

idealandj max1leilem

bij1113966 1113967 j isin J2 (12)

Furthermore with respect to the attributes Cj (j isin J2)

with the assessment values in the form of preference or-derings the grey relational coefficients between their crispvalues bij and the corresponding positive ideal attribute valueidealandj are defined as

qij miniminj dis2 bij ideal

andj1113872 11138731113966 1113967 + ρ maximaxj dis2 bij ideal

andj1113872 11138731113966 1113967

dis2 bij idealandj1113872 1113873 + ρ maximaxj dis2 bij ideal

andj1113872 11138731113966 1113967

i 1 m j isin J2 (13)

where dis2(middot) is the distance function between bij and idealandjand defined as follows

Dis2 bij idealandj1113872 1113873 idealandj minus bij i 1 m j isin J2

(14)

413 Calculate the Grey Correlation Coefficients of the In-terval Attribute Values With respect to the attributes Cj

(j isin J3) with interval assessment values denote 1113957aij as in-terval numbers [aL

ij aUij](i 1 m j isin J3) [aL

ij aUij] can

be transformed into the benefit type denoted as [bLij bU

ij]

(i 1 m j isin J3)

Definition 2 Given interval attribute values 1113957bij and 1113957bkj forCj i k 1 m j isin J3 the distance between 1113957bij and 1113957bkj isdefined as

dis31113957bij

1113957bkj1113872 1113873

2

radic

2

bLij minus bL

kj1113874 11138752

+ bUij minus bU

kj1113874 11138752

1113971

i k 1 m j isin J3 (15)

Definition 3 With respect to the attributes Cj (j isin J3) withinterval assessment values after the interval attribute values[aL

ij aUij](i 1 m j isin J3) are transformed into the benefit

type [bLij bU

ij] the positive ideal attribute value for attributeCj is defined as follows

ideallowastj ideallowastLj ideallowastU

j1113960 1113961 j isin J3 (16)

where

ideallowast Lj max

1leilemb

Lij1113966 1113967 j isin J3 (17a)

ideallowastUj max

1leilemb

Uij1113966 1113967 j isin J3 (17b)

Definition 4 Given the hybrid decision matrix 1113957A [1113957aij]mtimesnwith respect to the attributes Cj (j isin J3) with interval as-sessment values the grey relational coefficients between

4 Complexity

[bLij bU

ij] and the corresponding positive ideal attribute valueideallowastj are defined as

qij miniminj dis3 1113957bij ideal

lowastj1113872 11138731113966 1113967 + ρ maximaxj dis3 1113957bij ideal

lowastj1113872 11138731113966 1113967

dis3 1113957bij ideallowastj1113872 1113873 + ρ maximaxj dis3 1113957bij ideal

lowastj1113872 11138731113966 1113967

i 1 m j isin J3 (18)

where dis3(middot) is the distance function between two intervalnumbers as defined in (15) and ρ is the parameter whichusually has a value of 05

42 Determine Attribute Weights After the hybrid decisionmatrix 1113957A [1113957aij]mtimesn is transformed into Q [qij]mtimesn in thissection a deviation maximization model is proposed firstlyto determine the attribute weights and integrate them withthose derived by the entropy method

421 lte Proposed Deviation Maximization Model

Definition 5 Given the normalized and beneficial decisionmatrix Q (qij)mtimesn for attribute Cj the weighted distancebetween alternative Si and all other alternatives is defined as

Dij(W) 1113944m

k1dev qij qkj1113872 1113873wj i 1 m j 1 n

(19)

where W (w1 w2 wn) is the weight vector of the at-tributes and dev (middot) is the difference function between twoattribute values and is defined as follows

dev qij qkj1113872 1113873 qij minus qkj

11138681113868111386811138681113868

11138681113868111386811138681113868 i k 1 m j 1 n

(20)

Definition 6 Given the normalized and beneficial decisionmatrix Q (qij)mtimesn for attribute Cj the weighted distancebetween all alternatives and others is defined as

Dj(W) 1113944m

i1Dij(W) 1113944

m

i11113944

m

k1dev qij qkj1113872 1113873wj j 1 n

(21)

It can be seen that Dj(w) denotes the weighted distancesamong all alternatives for attribute Cj j 1 n As-suming that all the alternatives are equally competitive andthere is no preference between them the optimal weightvector W(W (w1 w2 wn)) of the attributes shouldmaximize the weighted distances among all alternativesacross all the attributes )erefore the following deviationmaximization model is set up to determine the attributeweights

maxD(w) 1113944n

j1Dj(w) 1113944

m

i11113944

n

j11113944

m

k1dev qij qkj1113872 1113873wj

(22a)

st

1113944

n

j1w

2j 1 j 1 n (22b)

wj ge 0 j 1 n (22c)

Theorem 1 lte optimal solution to model (22a)ndash(22c) is

wj 1113936

mi1 1113936

mk1 dev qij qkj1113872 1113873

1113936nj1 1113936

mi1 1113936

mk1 dev qij qkj1113872 11138731113872 1113873

21113969 j 1 n

(23)

Proof)e following Lagrange function is constructed

L(W λ) 1113944m

i11113944

n

j11113944

m

k1dev qij qkj1113872 1113873wj +

λ2

1113944

n

j1w

2j minus 1⎞⎠⎛⎝

(24)

where λ is the Lagrange multiplierLet zLzwj 0 and zLzλ 0 the following equation

can be obtained

1113944

m

i11113944

m

k1dev qij qkj1113872 1113873 + λwj 0 j 1 n (25a)

1113944

n

j1w

2j 1 (25b)

By solving the equations composed of (25a) and (25b)the following can be obtained

λ minus 1113944n

j11113944

m

i11113944

n

k1dev qij qkj1113872 1113873⎞⎠⎤⎦

12

⎛⎝⎡⎢⎢⎢⎢⎣ (26a)

wj 1113936

mi1 1113936

mk1 dev qij qkj1113872 1113873

1113936nj1 1113936

mi1 1113936

mk1 dev qij qkj1113872 11138731113872 1113873

21113969 k 1 m

(26b)

Complexity 5

Furthermore normalize the weights given by (26b) andthe weight of attribute Cj(j 1 n) is obtained

w1j

1113936mi1 1113936

mk1 dev qij qkj1113872 1113873

1113936nj1 1113936

mi1 1113936

mk1 dev qij qkj1113872 11138731113872 1113873

j 1 n (27)

)us weight vector W1 (W1 (w11 w1

2 w1n)) of the

attributes can be obtained based on maximizing theweighted distances among all alternatives across theattributes

422 Entropy Method Based on the normalized and ben-eficial decisionmatrix Q (qij)mtimesn the attribute weights canbe calculated by means of the entropy method [25]

(a) Calculate the proportions of attributes in matrix QGiven the normalized and beneficial decision matrixQ (qij)mtimesn for attribute Cj the ratio of qij to the sumof all elements in the same column in Q is calculated as

Zij qij

1113936nl1 qlj

i 1 m j 1 n (28)

(b) Calculate the information entropy of the attributes)e information entropy of the attributes can be cal-culated as follows

Ej minusk 1113944m

i1Zij ln zij j 1 n (29)

where k is the adjustment coefficient and k 1ln n(c) Calculate the redundancy of information entropy

ej 1 minus Ej j 1 n (30)

(d) Calculate the weights of attributes

Based on the information entropy of the attributes theirweights can be calculated as follows

Wj ej

1113936nj1 ej

j 1 n (31)

Denote W2 (w21 w2

2 w2n) as the weight vector of

the attributes obtained by means of the entropy method

423 Determine Comprehensive Attribute Weights Basedon the attribute weight vectorW1 calculated by the deviationmaximization model (22a)ndash(22c) and the attribute weightvector W2 obtained by the Entropy method the compre-hensive attribute weight vector based on the decision matrixQ is calculated in the following

Wo

05W1

+ 05W2 (32)

43 Calculate the Overall Values of Suppliers )e overallvalues of supplier Si can be obtained by means of theweighted sum method as follows

overalli 1113944n

j1w

ojqij i 1 m (33)

All the suppliers can be ranked descendingly accordingto their overall values calculated in formula (33)

5 Illustrations

In the course of supplier selection the determination ofassessment attributes is the first step )e selection of at-tributes is reviewed in [9] In this study service level (C1)degree of informatization (C2) profitability (C3) level ofquality (C4) and level of risk (C5) are adopted in evaluatingand selecting four suppliers (ie Si i 1 2 3 and 4) in B2Be-commerce environment Because of the uncertainty andfuzziness in B2B e-commerce environment for the attri-butes of service level (C1) and level of risk (C5) preferenceorderings are used

For the attributes of degree of informatization (C2) andlevel of quality (C4) linguistic terms are used to assess thesupplier performances For the attribute of profitability C3interval numbers are employed For the sake of simplicitythe linguistic term set ldquovery poorrdquo ldquopoorrdquo ldquofairrdquo ldquogoodrdquoand ldquovery goodrdquo is employed for both attributes of degree ofinformatization (C2) and level of quality (C4) and is same asthe basic linguistic term set TERMSETB in this study Detailsof the assessment information of the suppliers (Si i 1 2 34) against the attributes are stated in Table 1

Firstly with respect to attribute C1 of the service levelthe assessment information of the suppliers in Table 1 isnormalized as (b11 b21 b31 b41)T (1 0 03333 06667)Tandis further transformed as (q11 q21 q31 q41)T (1 0333304286 06)T by calculating the grey relational coefficientsbetween their crisp values bi1 (i 1 2 3 4) and the corre-sponding positive ideal attribute value

Secondly with respect to attribute C2 of degree of infor-matization the linguistic assessment information of the suppliersin Table 1 is normalized as (b12 b22 b32 b42)T (09167 025 05075)T and is further transformed as (q12 q22 q32 q42)T (1 0405714 1)T

)irdly with respect to attribute C3 of profitability theinterval assessment information of the suppliers in Table 1 isnormalized as

bl13 bu

13

bl23 bu

23

bl33 bu

33

bl43 bu

43

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

075 1

0 025

05 075

025 05

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

and it is further transformed as (q13 q23 q33 q43)T (0583605182 1 06940)T

Fourthly with respect to attribute C4 of level ofquality the linguistic assessment information of thesuppliers in Table 1 is normalized as (b14 b24 b34 b44)T (05 00833 075 09167)T and is further transformedas (q14 q24 q34 q44)T (05 03333 07143 1)T Fifthlywith respect to attribute C5 of level of risk the assessmentinformation of the suppliers in Table 1 is normalized into

6 Complexity

(b15 b25 b35 b45)T (06667 1 0 03333)T and is furthertransformed as (q15 q25 q35 q45)T (06 1 0333304286)T

Based on the above calculation the single-point valuedecision matrix Q is obtained as

Q

10000 10000 05836 05000 06000

03333 04000 05182 03333 10000

04286 05714 10000 07143 03333

06000 10000 06940 10000 04286

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (35)

Furthermore based on the normalized decision matrixQthe attribute weight vector can be calculated by the deviationmaximization model (22a)ndash(22c) as W1 (02100 0215501504 02141 and 02100) In the meantime the attributeweight vector can also be obtained by the entropy weightmethod as W2 (02120 01970 01746 02044 and 02120))us the comprehensive attribute weight vector based on thedecision matrix Q is calculated as Wo (02110 0206201625 02092 and 02110) Accordingly the overall values ofthe suppliers can be obtained as follows overall1 07432overall2 05177 overall3 06295 and overall4 07452 Fi-nally the ranking of the suppliers is S4gt S1gt S3gt S2

6 Conclusions

)is paper proposes an approach to evaluating and selectingthe manufacturing suppliers in B2B e-commerce environ-ment where linguistic terms preference orderings andinterval numbers are employed to present their fuzzy per-formances After the hybrid decision matrix is normalizedthe attribute weights are determined by means of proposingthe deviation maximization model and the entropy method

)e merits of the proposed approach lie in three aspects)e first one is to express the uncertainty of the suppliersrsquoperformances by means of the appropriate and the easiestways ie linguistic terms preference orderings and intervalnumbers )e second one is to propose the methods ofnormalizing the hybrid decision matrix by calculating thegrey correlation coefficients of attribute values with the idealvalues of attributes )e third one is to determine the at-tribute weights by means of the deviation maximizationmodel and the entropy method based on normalized de-cision matrix )is paper enables to express the suppliersrsquoperformance information in the easiest ways and accuratelyespecially in fuzzy or uncertain decision environmentCompared with the current research the proposed approachhas more universal significance and practical applicationprospects

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is work was supported by the Science and TechnologyAgency of Liaoning Province under the Grant of the NaturalScience Foundation of Liaoning Province (2013020022)ldquoHybrid multicriteria group decision-making with variousforms of information expressionrdquo

References

[1] D Maffin and P Braiden ldquoManufacturing and supplier rolesin product developmentrdquo International Journal of ProductionEconomics vol 69 no 2 pp 205ndash213 2001

[2] G Svensson T Mysen and J Payan ldquoBalancing the se-quential logic of quality constructs in manufacturing-supplierrelationshipsmdashcauses and outcomesrdquo Journal of BusinessResearch vol 63 no 11 pp 1209ndash1214 2010

[3] M J Saenz D Knoppen and E M Tachizawa ldquoBuildingmanufacturing flexibility with strategic suppliers and con-tingent effect of product dynamism on customer satisfactionrdquoJournal of Purchasing and Supply Management vol 24 no 3pp 238ndash246 2018

[4] T-C Chu and R Varma ldquoEvaluating suppliers via a multiplelevels multiple criteria decision making method under fuzzyenvironmentrdquo Computers amp Industrial Engineering vol 62no 2 pp 653ndash660 2012

[5] L Xie J Ma and H Han ldquoImplications of stochastic demandand manufacturersrsquo operational mode on retailerrsquos mixedbundling strategy and its complexity analysisrdquo AppliedMathematical Modelling vol 55 no 1 pp 484ndash501 2018

[6] J Ma and L Xie ldquo)e stability analysis of the dynamic pricingstrategy for bundling goods a comparison between simul-taneous and sequential pricing mechanismrdquo Nonlinear Dy-namics vol 95 no 2 pp 1147ndash1164 2019

[7] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computational Linguistics vol 9 no 1pp 149ndash184 1983

[8] L F D O M Santos L Osiro and R H P Lima ldquoA modelbased on 2-tuple fuzzy linguistic representation and AnalyticHierarchy Process for supplier segmentation using qualitativeand quantitative criteriardquo Expert Systems with Applicationsvol 79 pp 53ndash64 2017

[9] P Sureeyatanapas K Sriwattananusart T NiyamosothW Sessomboon and S Arunyanart ldquoSupplier selection to-wards uncertain and unavailable information an extension of

Table 1 )e assessment information of the suppliers

Service level C1 Degree of informatization C2 Profitability C3 Level of quality C4 Level of risk C5

S1 1 Poor [70 80] Fair 2S2 4 Very poor [40 50] Poor 1S3 3 Fair [60 70] Very good 4S4 2 Very good [50 60] Good 3

Complexity 7

TOPSIS methodrdquo Operations Research Perspectives vol 5pp 69ndash79 2018

[10] C-N Liao and H-P Kao ldquoAn integrated fuzzy TOPSIS andMCGP approach to supplier selection in supply chainmanagementrdquo Expert Systems with Applications vol 38 no 9pp 10803ndash10811 2011

[11] F R L Junior L Osiro and L C R Carpinetti ldquoA com-parison between fuzzy AHP and fuzzy TOPSIS methods tosupplier selectionrdquo Applied Soft Computing vol 21 no 5pp 194ndash209 2014

[12] A Azizi D O Aikhuele and F S Souleman ldquoA fuzzy TOPSISmodel to rank automotive suppliersrdquo Procedia Manufactur-ing vol 2 pp 159ndash164 2015

[13] A Kokangul and Z Susuz ldquoIntegrated analytical hierarchprocess and mathematical programming to supplier selectionproblem with quantity discountrdquo Applied MathematicalModelling vol 33 no 3 pp 1417ndash1429 2009

[14] N R Ware S P Singh and D K Banwet ldquoA mixed-integernon-linear program to model dynamic supplier selectionproblemrdquo Expert Systems with Applications vol 41 no 2pp 671ndash678 2014

[15] H Mirzaee B Naderi and S H R Pasandideh ldquoA pre-emptive fuzzy goal programming model for generalizedsupplier selection and order allocation with incrementaldiscountrdquo Computers amp Industrial Engineering vol 122pp 292ndash302 2018

[16] C L Hwang and K Yoon Multiple Attribute DecisionMaking Methods and Applications Springer-Verlag NewYork NY USA 1981

[17] C C Lee and C Ou-Yang ldquoA neural networks approach forforecasting the supplierrsquos bid prices in supplier selectionnegotiation processrdquo Expert Systems with Applicationsvol 36 no 2 pp 2961ndash2970 2009

[18] A F Guneri T Ertay and A Yucel ldquoAn approach based onANFIS input selection and modeling for supplier selectionproblemrdquo Expert Systems with Applications vol 38 no 12pp 14907ndash14917 2011

[19] C Bai and J Sarkis ldquoIntegrating sustainability into supplierselection with grey system and rough set methodologiesrdquoInternational Journal of Production Economics vol 124 no 1pp 252ndash264 2010

[20] D Golmohammadi and M Mellat-Parast ldquoDeveloping agrey-based decision-making model for supplier selectionrdquoInternational Journal of Production Economics vol 137 no 2pp 191ndash200 2012

[21] M S Memon Y H Lee and S I Mari ldquoGroup multi-criteriasupplier selection using combined grey systems theory anduncertainty theoryrdquo Expert Systems with Applications vol 42no 21 pp 7951ndash7959 2015

[22] B S Ahn and K S Park ldquoComparing methods for multi-attribute decisionmaking with ordinal weightsrdquo Computers ampOperations Research vol 35 no 5 pp 1660ndash1670 2008

[23] J Wang and S Zionts ldquoUsing ordinal data to estimate car-dinal valuesrdquo Journal of Multi-Criteria Decision Analysisvol 22 no 3ndash4 pp 185ndash196 2015

[24] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Eu-ropean Journal of Operational Research vol 166 no 1pp 115ndash132 2005

[25] Y Ji G H Huang and W Sun ldquoRisk assessment of hy-dropower stations through an integrated fuzzy entropy-weight multiple criteria decision making method a case studyof the Xiangxi Riverrdquo Expert Systems with Applicationsvol 42 no 12 pp 5380ndash5389 2015

8 Complexity

and assumptions are used to represent the MADM problemof evaluating and selecting manufacturing suppliers

)e alternatives (ie the suppliers) are known Let S

S1 S2 Sm1113864 1113865 denote a discrete set of m(ge2) possible al-ternatives )e attributes are known and letC C1 C2 Cn1113864 1113865 denote a set of n(ge2) attributes Inorder to distinguish the qualitative and quantitative attri-butes with different characteristics the subscript of the at-tributes is divided into three categories J1 for the subscriptset of attributes with linguistic assessment values J2 for thesubscript set of attributes with preference ordering assess-ment values and J3 for the subscript set of attributes withinterval number assessment values

Let 1113957A [1113957aij]mtimesn denote the decision matrix where 1113957aij

are the assessment values for alternative Si with respect toattribute Cj i 1 m and j 1 n In this study 1113957aij

are in the forms of linguistic terms preference orderingsand interval numbers

Let W (w1 w2 wn) denote the weight vector ofn(ge 2) attributes where wj is the weight of attribute Cjwhile 1113936

nj1 wj 1 and wj gt 0 holds for j 1 n

)e problem focused in this paper is to select the bestsupplier(s) for a manufacturer in B2B E-commerce envi-ronment while their performance indicators (ie attributevalues) are linguistic terms preference orderings and in-terval numbers

4 The Proposed Approach

)e proposed approach to the problem stated in Section 3 iscomposed of three steps normalize the attribute values indifferent formats based on the grey relational degreemethod determine the attribute weights and calculate theoverall values of the alternatives (suppliers)

41 Normalize the Attribute Values in Different FormatsSince the attribute values in decision matrix 1113957A [1113957aij]mtimesn arein the formats of linguistic terms preference orderings andinterval numbers corresponding methods are developed fortransforming them into a comparable format ie the utilityvalue

411 Calculate the Grey Correlation Coefficients of LinguisticAttribute Values

Definition 1 A linguistic term 1113957T on a real-number set isdefined as a triangular fuzzy number (denoted as (u α andβ)) if its membership function μ1113957T

(R+⟶ [0 1]) is definedas

μ1113957T(x)

x minus αu minus α

x isin [α u]

x minus βu minus β

x isin [u β]

0 otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

where αle ule β u is the model value and α and β stand forthe lower value and the upper value of linguistic term 1113957Trespectively

Given the hybrid decision matrix 1113957A [1113957aij]mtimesn the at-tributes Cj (j isin J1) with linguistic assessment values may beof different granularities and different linguistic evaluationsets would be employed [24] )erefore a basic linguisticevaluation set TERMSETB (TERMSETB termB

0 termB1

termBg) is used to transform the linguistic assessment values

with different granularities into the comparable form )etriangular fuzzy number cB

l (μBl αB

l βBl ) corresponding to

the linguistic term termBi is defined as follows

cBl

αB0 0

uBl

l

g minus 1 0le lleg minus 1

αBl

l minus 1g minus 1

1le lleg minus 1

βBl

l + 1g minus 1

0le lleg minus 2

βBgminus1 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)

where μBl is the model value and αB

l and βBl stand for the

lower value and the upper value of cBl

With respect to the attribute values with linguistic as-sessments 1113957aij (i 1 m j isin J1) denote 1113957aij as lingij lingij

can be transformed into the fuzzy set over the basic linguisticevaluation set TERMSETB (denoted as Fij (TERMSETB))

τ lingij⟶ Fij TERMSETB1113872 1113873 i 1 m j isin J1

(3)

where Fij(TERMSETB) is the fuzzy set over the basic lin-guistic evaluation set TERMSETB as stated as follows

Fij TERMSETB1113872 1113873 termB

l cijl1113872 1113873|l isin [0 g] l 0 g1113966 1113967

(4)

where

cijl maxy

min μlingij(y) μtermB

l(y)1113882 1113883 (5)

where μlingij(y) and μtermB

l(y) denote the membership

functions of lingij and termBl respectively i 1 m l 0

g j isin J1Furthermore the fuzzy set Fij (TERMSETB) can be

transformed into a crisp value as follows

ϕ Fij TERMSETB1113872 11138731113872 1113873

1113936g

l0 l times cijl

g 1113936g

l0 cijl

i 1 m j isin J1

(6)

)us given the linguistic assessment values 1113957aij of at-tributes Cj (j isin J1) by means of the operations in (3)ndash(6)

Complexity 3

linguistic term 1113957aij is transformed into the form of crispvalue denoted as

bij ϕ Fij TERMSETB1113872 11138731113872 1113873 i 1 m j isin J1 (7)

In the meantime the positive ideal attribute value forattribute Cj (j isin J1) is defined as follows

ideal+j max1leilem

bij1113966 1113967 j isin J1 (8)

)erefore with respect to the attributes Cj (j isin J1) withlinguistic assessment values the grey relational coefficientsbetween their crisp values bij and the corresponding positiveideal attribute value ideal+j are defined as

qij miniminj dis bij ideal

+j1113872 11138731113966 1113967 + ρ maximaxj dis1 bij ideal

+j1113872 11138731113966 1113967

dis1 bij ideal+j1113872 1113873 + ρ maximaxj dis1 bij ideal

+j1113872 11138731113966 1113967

i 1 m j isin J1 (9)

where dis1(middot) is the distance function between bij and ideal+jand is defined as follows

Dis1 bij ideal+j1113872 1113873 ideal+j minus bij i 1 m j isin J1

(10)

412 Calculate the Grey Correlation Coefficients of the At-tribute Values in the Form of Preference Orderings Withrespect to the attributes Cj (j isin J2) with the assessmentvalues in the form of preference orderings denote 1113957aij(i 1 m j isin J2) as Rij and Rij is a permutation function overthe index set 1 m Alternatively Rij represents theposition of Si in the preference ordering Rij can be trans-formed into a crisp value bij as follows

bij m minus Rij

m minus 1 i 1 m j isin J2 (11)

In addition regarding the attributes Cj (j isin J2) with theassessment values in the form of preference orderings thepositive ideal attribute value for attribute Cj is defined asfollows

idealandj max1leilem

bij1113966 1113967 j isin J2 (12)

Furthermore with respect to the attributes Cj (j isin J2)

with the assessment values in the form of preference or-derings the grey relational coefficients between their crispvalues bij and the corresponding positive ideal attribute valueidealandj are defined as

qij miniminj dis2 bij ideal

andj1113872 11138731113966 1113967 + ρ maximaxj dis2 bij ideal

andj1113872 11138731113966 1113967

dis2 bij idealandj1113872 1113873 + ρ maximaxj dis2 bij ideal

andj1113872 11138731113966 1113967

i 1 m j isin J2 (13)

where dis2(middot) is the distance function between bij and idealandjand defined as follows

Dis2 bij idealandj1113872 1113873 idealandj minus bij i 1 m j isin J2

(14)

413 Calculate the Grey Correlation Coefficients of the In-terval Attribute Values With respect to the attributes Cj

(j isin J3) with interval assessment values denote 1113957aij as in-terval numbers [aL

ij aUij](i 1 m j isin J3) [aL

ij aUij] can

be transformed into the benefit type denoted as [bLij bU

ij]

(i 1 m j isin J3)

Definition 2 Given interval attribute values 1113957bij and 1113957bkj forCj i k 1 m j isin J3 the distance between 1113957bij and 1113957bkj isdefined as

dis31113957bij

1113957bkj1113872 1113873

2

radic

2

bLij minus bL

kj1113874 11138752

+ bUij minus bU

kj1113874 11138752

1113971

i k 1 m j isin J3 (15)

Definition 3 With respect to the attributes Cj (j isin J3) withinterval assessment values after the interval attribute values[aL

ij aUij](i 1 m j isin J3) are transformed into the benefit

type [bLij bU

ij] the positive ideal attribute value for attributeCj is defined as follows

ideallowastj ideallowastLj ideallowastU

j1113960 1113961 j isin J3 (16)

where

ideallowast Lj max

1leilemb

Lij1113966 1113967 j isin J3 (17a)

ideallowastUj max

1leilemb

Uij1113966 1113967 j isin J3 (17b)

Definition 4 Given the hybrid decision matrix 1113957A [1113957aij]mtimesnwith respect to the attributes Cj (j isin J3) with interval as-sessment values the grey relational coefficients between

4 Complexity

[bLij bU

ij] and the corresponding positive ideal attribute valueideallowastj are defined as

qij miniminj dis3 1113957bij ideal

lowastj1113872 11138731113966 1113967 + ρ maximaxj dis3 1113957bij ideal

lowastj1113872 11138731113966 1113967

dis3 1113957bij ideallowastj1113872 1113873 + ρ maximaxj dis3 1113957bij ideal

lowastj1113872 11138731113966 1113967

i 1 m j isin J3 (18)

where dis3(middot) is the distance function between two intervalnumbers as defined in (15) and ρ is the parameter whichusually has a value of 05

42 Determine Attribute Weights After the hybrid decisionmatrix 1113957A [1113957aij]mtimesn is transformed into Q [qij]mtimesn in thissection a deviation maximization model is proposed firstlyto determine the attribute weights and integrate them withthose derived by the entropy method

421 lte Proposed Deviation Maximization Model

Definition 5 Given the normalized and beneficial decisionmatrix Q (qij)mtimesn for attribute Cj the weighted distancebetween alternative Si and all other alternatives is defined as

Dij(W) 1113944m

k1dev qij qkj1113872 1113873wj i 1 m j 1 n

(19)

where W (w1 w2 wn) is the weight vector of the at-tributes and dev (middot) is the difference function between twoattribute values and is defined as follows

dev qij qkj1113872 1113873 qij minus qkj

11138681113868111386811138681113868

11138681113868111386811138681113868 i k 1 m j 1 n

(20)

Definition 6 Given the normalized and beneficial decisionmatrix Q (qij)mtimesn for attribute Cj the weighted distancebetween all alternatives and others is defined as

Dj(W) 1113944m

i1Dij(W) 1113944

m

i11113944

m

k1dev qij qkj1113872 1113873wj j 1 n

(21)

It can be seen that Dj(w) denotes the weighted distancesamong all alternatives for attribute Cj j 1 n As-suming that all the alternatives are equally competitive andthere is no preference between them the optimal weightvector W(W (w1 w2 wn)) of the attributes shouldmaximize the weighted distances among all alternativesacross all the attributes )erefore the following deviationmaximization model is set up to determine the attributeweights

maxD(w) 1113944n

j1Dj(w) 1113944

m

i11113944

n

j11113944

m

k1dev qij qkj1113872 1113873wj

(22a)

st

1113944

n

j1w

2j 1 j 1 n (22b)

wj ge 0 j 1 n (22c)

Theorem 1 lte optimal solution to model (22a)ndash(22c) is

wj 1113936

mi1 1113936

mk1 dev qij qkj1113872 1113873

1113936nj1 1113936

mi1 1113936

mk1 dev qij qkj1113872 11138731113872 1113873

21113969 j 1 n

(23)

Proof)e following Lagrange function is constructed

L(W λ) 1113944m

i11113944

n

j11113944

m

k1dev qij qkj1113872 1113873wj +

λ2

1113944

n

j1w

2j minus 1⎞⎠⎛⎝

(24)

where λ is the Lagrange multiplierLet zLzwj 0 and zLzλ 0 the following equation

can be obtained

1113944

m

i11113944

m

k1dev qij qkj1113872 1113873 + λwj 0 j 1 n (25a)

1113944

n

j1w

2j 1 (25b)

By solving the equations composed of (25a) and (25b)the following can be obtained

λ minus 1113944n

j11113944

m

i11113944

n

k1dev qij qkj1113872 1113873⎞⎠⎤⎦

12

⎛⎝⎡⎢⎢⎢⎢⎣ (26a)

wj 1113936

mi1 1113936

mk1 dev qij qkj1113872 1113873

1113936nj1 1113936

mi1 1113936

mk1 dev qij qkj1113872 11138731113872 1113873

21113969 k 1 m

(26b)

Complexity 5

Furthermore normalize the weights given by (26b) andthe weight of attribute Cj(j 1 n) is obtained

w1j

1113936mi1 1113936

mk1 dev qij qkj1113872 1113873

1113936nj1 1113936

mi1 1113936

mk1 dev qij qkj1113872 11138731113872 1113873

j 1 n (27)

)us weight vector W1 (W1 (w11 w1

2 w1n)) of the

attributes can be obtained based on maximizing theweighted distances among all alternatives across theattributes

422 Entropy Method Based on the normalized and ben-eficial decisionmatrix Q (qij)mtimesn the attribute weights canbe calculated by means of the entropy method [25]

(a) Calculate the proportions of attributes in matrix QGiven the normalized and beneficial decision matrixQ (qij)mtimesn for attribute Cj the ratio of qij to the sumof all elements in the same column in Q is calculated as

Zij qij

1113936nl1 qlj

i 1 m j 1 n (28)

(b) Calculate the information entropy of the attributes)e information entropy of the attributes can be cal-culated as follows

Ej minusk 1113944m

i1Zij ln zij j 1 n (29)

where k is the adjustment coefficient and k 1ln n(c) Calculate the redundancy of information entropy

ej 1 minus Ej j 1 n (30)

(d) Calculate the weights of attributes

Based on the information entropy of the attributes theirweights can be calculated as follows

Wj ej

1113936nj1 ej

j 1 n (31)

Denote W2 (w21 w2

2 w2n) as the weight vector of

the attributes obtained by means of the entropy method

423 Determine Comprehensive Attribute Weights Basedon the attribute weight vectorW1 calculated by the deviationmaximization model (22a)ndash(22c) and the attribute weightvector W2 obtained by the Entropy method the compre-hensive attribute weight vector based on the decision matrixQ is calculated in the following

Wo

05W1

+ 05W2 (32)

43 Calculate the Overall Values of Suppliers )e overallvalues of supplier Si can be obtained by means of theweighted sum method as follows

overalli 1113944n

j1w

ojqij i 1 m (33)

All the suppliers can be ranked descendingly accordingto their overall values calculated in formula (33)

5 Illustrations

In the course of supplier selection the determination ofassessment attributes is the first step )e selection of at-tributes is reviewed in [9] In this study service level (C1)degree of informatization (C2) profitability (C3) level ofquality (C4) and level of risk (C5) are adopted in evaluatingand selecting four suppliers (ie Si i 1 2 3 and 4) in B2Be-commerce environment Because of the uncertainty andfuzziness in B2B e-commerce environment for the attri-butes of service level (C1) and level of risk (C5) preferenceorderings are used

For the attributes of degree of informatization (C2) andlevel of quality (C4) linguistic terms are used to assess thesupplier performances For the attribute of profitability C3interval numbers are employed For the sake of simplicitythe linguistic term set ldquovery poorrdquo ldquopoorrdquo ldquofairrdquo ldquogoodrdquoand ldquovery goodrdquo is employed for both attributes of degree ofinformatization (C2) and level of quality (C4) and is same asthe basic linguistic term set TERMSETB in this study Detailsof the assessment information of the suppliers (Si i 1 2 34) against the attributes are stated in Table 1

Firstly with respect to attribute C1 of the service levelthe assessment information of the suppliers in Table 1 isnormalized as (b11 b21 b31 b41)T (1 0 03333 06667)Tandis further transformed as (q11 q21 q31 q41)T (1 0333304286 06)T by calculating the grey relational coefficientsbetween their crisp values bi1 (i 1 2 3 4) and the corre-sponding positive ideal attribute value

Secondly with respect to attribute C2 of degree of infor-matization the linguistic assessment information of the suppliersin Table 1 is normalized as (b12 b22 b32 b42)T (09167 025 05075)T and is further transformed as (q12 q22 q32 q42)T (1 0405714 1)T

)irdly with respect to attribute C3 of profitability theinterval assessment information of the suppliers in Table 1 isnormalized as

bl13 bu

13

bl23 bu

23

bl33 bu

33

bl43 bu

43

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

075 1

0 025

05 075

025 05

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

and it is further transformed as (q13 q23 q33 q43)T (0583605182 1 06940)T

Fourthly with respect to attribute C4 of level ofquality the linguistic assessment information of thesuppliers in Table 1 is normalized as (b14 b24 b34 b44)T (05 00833 075 09167)T and is further transformedas (q14 q24 q34 q44)T (05 03333 07143 1)T Fifthlywith respect to attribute C5 of level of risk the assessmentinformation of the suppliers in Table 1 is normalized into

6 Complexity

(b15 b25 b35 b45)T (06667 1 0 03333)T and is furthertransformed as (q15 q25 q35 q45)T (06 1 0333304286)T

Based on the above calculation the single-point valuedecision matrix Q is obtained as

Q

10000 10000 05836 05000 06000

03333 04000 05182 03333 10000

04286 05714 10000 07143 03333

06000 10000 06940 10000 04286

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (35)

Furthermore based on the normalized decision matrixQthe attribute weight vector can be calculated by the deviationmaximization model (22a)ndash(22c) as W1 (02100 0215501504 02141 and 02100) In the meantime the attributeweight vector can also be obtained by the entropy weightmethod as W2 (02120 01970 01746 02044 and 02120))us the comprehensive attribute weight vector based on thedecision matrix Q is calculated as Wo (02110 0206201625 02092 and 02110) Accordingly the overall values ofthe suppliers can be obtained as follows overall1 07432overall2 05177 overall3 06295 and overall4 07452 Fi-nally the ranking of the suppliers is S4gt S1gt S3gt S2

6 Conclusions

)is paper proposes an approach to evaluating and selectingthe manufacturing suppliers in B2B e-commerce environ-ment where linguistic terms preference orderings andinterval numbers are employed to present their fuzzy per-formances After the hybrid decision matrix is normalizedthe attribute weights are determined by means of proposingthe deviation maximization model and the entropy method

)e merits of the proposed approach lie in three aspects)e first one is to express the uncertainty of the suppliersrsquoperformances by means of the appropriate and the easiestways ie linguistic terms preference orderings and intervalnumbers )e second one is to propose the methods ofnormalizing the hybrid decision matrix by calculating thegrey correlation coefficients of attribute values with the idealvalues of attributes )e third one is to determine the at-tribute weights by means of the deviation maximizationmodel and the entropy method based on normalized de-cision matrix )is paper enables to express the suppliersrsquoperformance information in the easiest ways and accuratelyespecially in fuzzy or uncertain decision environmentCompared with the current research the proposed approachhas more universal significance and practical applicationprospects

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is work was supported by the Science and TechnologyAgency of Liaoning Province under the Grant of the NaturalScience Foundation of Liaoning Province (2013020022)ldquoHybrid multicriteria group decision-making with variousforms of information expressionrdquo

References

[1] D Maffin and P Braiden ldquoManufacturing and supplier rolesin product developmentrdquo International Journal of ProductionEconomics vol 69 no 2 pp 205ndash213 2001

[2] G Svensson T Mysen and J Payan ldquoBalancing the se-quential logic of quality constructs in manufacturing-supplierrelationshipsmdashcauses and outcomesrdquo Journal of BusinessResearch vol 63 no 11 pp 1209ndash1214 2010

[3] M J Saenz D Knoppen and E M Tachizawa ldquoBuildingmanufacturing flexibility with strategic suppliers and con-tingent effect of product dynamism on customer satisfactionrdquoJournal of Purchasing and Supply Management vol 24 no 3pp 238ndash246 2018

[4] T-C Chu and R Varma ldquoEvaluating suppliers via a multiplelevels multiple criteria decision making method under fuzzyenvironmentrdquo Computers amp Industrial Engineering vol 62no 2 pp 653ndash660 2012

[5] L Xie J Ma and H Han ldquoImplications of stochastic demandand manufacturersrsquo operational mode on retailerrsquos mixedbundling strategy and its complexity analysisrdquo AppliedMathematical Modelling vol 55 no 1 pp 484ndash501 2018

[6] J Ma and L Xie ldquo)e stability analysis of the dynamic pricingstrategy for bundling goods a comparison between simul-taneous and sequential pricing mechanismrdquo Nonlinear Dy-namics vol 95 no 2 pp 1147ndash1164 2019

[7] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computational Linguistics vol 9 no 1pp 149ndash184 1983

[8] L F D O M Santos L Osiro and R H P Lima ldquoA modelbased on 2-tuple fuzzy linguistic representation and AnalyticHierarchy Process for supplier segmentation using qualitativeand quantitative criteriardquo Expert Systems with Applicationsvol 79 pp 53ndash64 2017

[9] P Sureeyatanapas K Sriwattananusart T NiyamosothW Sessomboon and S Arunyanart ldquoSupplier selection to-wards uncertain and unavailable information an extension of

Table 1 )e assessment information of the suppliers

Service level C1 Degree of informatization C2 Profitability C3 Level of quality C4 Level of risk C5

S1 1 Poor [70 80] Fair 2S2 4 Very poor [40 50] Poor 1S3 3 Fair [60 70] Very good 4S4 2 Very good [50 60] Good 3

Complexity 7

TOPSIS methodrdquo Operations Research Perspectives vol 5pp 69ndash79 2018

[10] C-N Liao and H-P Kao ldquoAn integrated fuzzy TOPSIS andMCGP approach to supplier selection in supply chainmanagementrdquo Expert Systems with Applications vol 38 no 9pp 10803ndash10811 2011

[11] F R L Junior L Osiro and L C R Carpinetti ldquoA com-parison between fuzzy AHP and fuzzy TOPSIS methods tosupplier selectionrdquo Applied Soft Computing vol 21 no 5pp 194ndash209 2014

[12] A Azizi D O Aikhuele and F S Souleman ldquoA fuzzy TOPSISmodel to rank automotive suppliersrdquo Procedia Manufactur-ing vol 2 pp 159ndash164 2015

[13] A Kokangul and Z Susuz ldquoIntegrated analytical hierarchprocess and mathematical programming to supplier selectionproblem with quantity discountrdquo Applied MathematicalModelling vol 33 no 3 pp 1417ndash1429 2009

[14] N R Ware S P Singh and D K Banwet ldquoA mixed-integernon-linear program to model dynamic supplier selectionproblemrdquo Expert Systems with Applications vol 41 no 2pp 671ndash678 2014

[15] H Mirzaee B Naderi and S H R Pasandideh ldquoA pre-emptive fuzzy goal programming model for generalizedsupplier selection and order allocation with incrementaldiscountrdquo Computers amp Industrial Engineering vol 122pp 292ndash302 2018

[16] C L Hwang and K Yoon Multiple Attribute DecisionMaking Methods and Applications Springer-Verlag NewYork NY USA 1981

[17] C C Lee and C Ou-Yang ldquoA neural networks approach forforecasting the supplierrsquos bid prices in supplier selectionnegotiation processrdquo Expert Systems with Applicationsvol 36 no 2 pp 2961ndash2970 2009

[18] A F Guneri T Ertay and A Yucel ldquoAn approach based onANFIS input selection and modeling for supplier selectionproblemrdquo Expert Systems with Applications vol 38 no 12pp 14907ndash14917 2011

[19] C Bai and J Sarkis ldquoIntegrating sustainability into supplierselection with grey system and rough set methodologiesrdquoInternational Journal of Production Economics vol 124 no 1pp 252ndash264 2010

[20] D Golmohammadi and M Mellat-Parast ldquoDeveloping agrey-based decision-making model for supplier selectionrdquoInternational Journal of Production Economics vol 137 no 2pp 191ndash200 2012

[21] M S Memon Y H Lee and S I Mari ldquoGroup multi-criteriasupplier selection using combined grey systems theory anduncertainty theoryrdquo Expert Systems with Applications vol 42no 21 pp 7951ndash7959 2015

[22] B S Ahn and K S Park ldquoComparing methods for multi-attribute decisionmaking with ordinal weightsrdquo Computers ampOperations Research vol 35 no 5 pp 1660ndash1670 2008

[23] J Wang and S Zionts ldquoUsing ordinal data to estimate car-dinal valuesrdquo Journal of Multi-Criteria Decision Analysisvol 22 no 3ndash4 pp 185ndash196 2015

[24] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Eu-ropean Journal of Operational Research vol 166 no 1pp 115ndash132 2005

[25] Y Ji G H Huang and W Sun ldquoRisk assessment of hy-dropower stations through an integrated fuzzy entropy-weight multiple criteria decision making method a case studyof the Xiangxi Riverrdquo Expert Systems with Applicationsvol 42 no 12 pp 5380ndash5389 2015

8 Complexity

linguistic term 1113957aij is transformed into the form of crispvalue denoted as

bij ϕ Fij TERMSETB1113872 11138731113872 1113873 i 1 m j isin J1 (7)

In the meantime the positive ideal attribute value forattribute Cj (j isin J1) is defined as follows

ideal+j max1leilem

bij1113966 1113967 j isin J1 (8)

)erefore with respect to the attributes Cj (j isin J1) withlinguistic assessment values the grey relational coefficientsbetween their crisp values bij and the corresponding positiveideal attribute value ideal+j are defined as

qij miniminj dis bij ideal

+j1113872 11138731113966 1113967 + ρ maximaxj dis1 bij ideal

+j1113872 11138731113966 1113967

dis1 bij ideal+j1113872 1113873 + ρ maximaxj dis1 bij ideal

+j1113872 11138731113966 1113967

i 1 m j isin J1 (9)

where dis1(middot) is the distance function between bij and ideal+jand is defined as follows

Dis1 bij ideal+j1113872 1113873 ideal+j minus bij i 1 m j isin J1

(10)

412 Calculate the Grey Correlation Coefficients of the At-tribute Values in the Form of Preference Orderings Withrespect to the attributes Cj (j isin J2) with the assessmentvalues in the form of preference orderings denote 1113957aij(i 1 m j isin J2) as Rij and Rij is a permutation function overthe index set 1 m Alternatively Rij represents theposition of Si in the preference ordering Rij can be trans-formed into a crisp value bij as follows

bij m minus Rij

m minus 1 i 1 m j isin J2 (11)

In addition regarding the attributes Cj (j isin J2) with theassessment values in the form of preference orderings thepositive ideal attribute value for attribute Cj is defined asfollows

idealandj max1leilem

bij1113966 1113967 j isin J2 (12)

Furthermore with respect to the attributes Cj (j isin J2)

with the assessment values in the form of preference or-derings the grey relational coefficients between their crispvalues bij and the corresponding positive ideal attribute valueidealandj are defined as

qij miniminj dis2 bij ideal

andj1113872 11138731113966 1113967 + ρ maximaxj dis2 bij ideal

andj1113872 11138731113966 1113967

dis2 bij idealandj1113872 1113873 + ρ maximaxj dis2 bij ideal

andj1113872 11138731113966 1113967

i 1 m j isin J2 (13)

where dis2(middot) is the distance function between bij and idealandjand defined as follows

Dis2 bij idealandj1113872 1113873 idealandj minus bij i 1 m j isin J2

(14)

413 Calculate the Grey Correlation Coefficients of the In-terval Attribute Values With respect to the attributes Cj

(j isin J3) with interval assessment values denote 1113957aij as in-terval numbers [aL

ij aUij](i 1 m j isin J3) [aL

ij aUij] can

be transformed into the benefit type denoted as [bLij bU

ij]

(i 1 m j isin J3)

Definition 2 Given interval attribute values 1113957bij and 1113957bkj forCj i k 1 m j isin J3 the distance between 1113957bij and 1113957bkj isdefined as

dis31113957bij

1113957bkj1113872 1113873

2

radic

2

bLij minus bL

kj1113874 11138752

+ bUij minus bU

kj1113874 11138752

1113971

i k 1 m j isin J3 (15)

Definition 3 With respect to the attributes Cj (j isin J3) withinterval assessment values after the interval attribute values[aL

ij aUij](i 1 m j isin J3) are transformed into the benefit

type [bLij bU

ij] the positive ideal attribute value for attributeCj is defined as follows

ideallowastj ideallowastLj ideallowastU

j1113960 1113961 j isin J3 (16)

where

ideallowast Lj max

1leilemb

Lij1113966 1113967 j isin J3 (17a)

ideallowastUj max

1leilemb

Uij1113966 1113967 j isin J3 (17b)

Definition 4 Given the hybrid decision matrix 1113957A [1113957aij]mtimesnwith respect to the attributes Cj (j isin J3) with interval as-sessment values the grey relational coefficients between

4 Complexity

[bLij bU

ij] and the corresponding positive ideal attribute valueideallowastj are defined as

qij miniminj dis3 1113957bij ideal

lowastj1113872 11138731113966 1113967 + ρ maximaxj dis3 1113957bij ideal

lowastj1113872 11138731113966 1113967

dis3 1113957bij ideallowastj1113872 1113873 + ρ maximaxj dis3 1113957bij ideal

lowastj1113872 11138731113966 1113967

i 1 m j isin J3 (18)

where dis3(middot) is the distance function between two intervalnumbers as defined in (15) and ρ is the parameter whichusually has a value of 05

42 Determine Attribute Weights After the hybrid decisionmatrix 1113957A [1113957aij]mtimesn is transformed into Q [qij]mtimesn in thissection a deviation maximization model is proposed firstlyto determine the attribute weights and integrate them withthose derived by the entropy method

421 lte Proposed Deviation Maximization Model

Definition 5 Given the normalized and beneficial decisionmatrix Q (qij)mtimesn for attribute Cj the weighted distancebetween alternative Si and all other alternatives is defined as

Dij(W) 1113944m

k1dev qij qkj1113872 1113873wj i 1 m j 1 n

(19)

where W (w1 w2 wn) is the weight vector of the at-tributes and dev (middot) is the difference function between twoattribute values and is defined as follows

dev qij qkj1113872 1113873 qij minus qkj

11138681113868111386811138681113868

11138681113868111386811138681113868 i k 1 m j 1 n

(20)

Definition 6 Given the normalized and beneficial decisionmatrix Q (qij)mtimesn for attribute Cj the weighted distancebetween all alternatives and others is defined as

Dj(W) 1113944m

i1Dij(W) 1113944

m

i11113944

m

k1dev qij qkj1113872 1113873wj j 1 n

(21)

It can be seen that Dj(w) denotes the weighted distancesamong all alternatives for attribute Cj j 1 n As-suming that all the alternatives are equally competitive andthere is no preference between them the optimal weightvector W(W (w1 w2 wn)) of the attributes shouldmaximize the weighted distances among all alternativesacross all the attributes )erefore the following deviationmaximization model is set up to determine the attributeweights

maxD(w) 1113944n

j1Dj(w) 1113944

m

i11113944

n

j11113944

m

k1dev qij qkj1113872 1113873wj

(22a)

st

1113944

n

j1w

2j 1 j 1 n (22b)

wj ge 0 j 1 n (22c)

Theorem 1 lte optimal solution to model (22a)ndash(22c) is

wj 1113936

mi1 1113936

mk1 dev qij qkj1113872 1113873

1113936nj1 1113936

mi1 1113936

mk1 dev qij qkj1113872 11138731113872 1113873

21113969 j 1 n

(23)

Proof)e following Lagrange function is constructed

L(W λ) 1113944m

i11113944

n

j11113944

m

k1dev qij qkj1113872 1113873wj +

λ2

1113944

n

j1w

2j minus 1⎞⎠⎛⎝

(24)

where λ is the Lagrange multiplierLet zLzwj 0 and zLzλ 0 the following equation

can be obtained

1113944

m

i11113944

m

k1dev qij qkj1113872 1113873 + λwj 0 j 1 n (25a)

1113944

n

j1w

2j 1 (25b)

By solving the equations composed of (25a) and (25b)the following can be obtained

λ minus 1113944n

j11113944

m

i11113944

n

k1dev qij qkj1113872 1113873⎞⎠⎤⎦

12

⎛⎝⎡⎢⎢⎢⎢⎣ (26a)

wj 1113936

mi1 1113936

mk1 dev qij qkj1113872 1113873

1113936nj1 1113936

mi1 1113936

mk1 dev qij qkj1113872 11138731113872 1113873

21113969 k 1 m

(26b)

Complexity 5

Furthermore normalize the weights given by (26b) andthe weight of attribute Cj(j 1 n) is obtained

w1j

1113936mi1 1113936

mk1 dev qij qkj1113872 1113873

1113936nj1 1113936

mi1 1113936

mk1 dev qij qkj1113872 11138731113872 1113873

j 1 n (27)

)us weight vector W1 (W1 (w11 w1

2 w1n)) of the

attributes can be obtained based on maximizing theweighted distances among all alternatives across theattributes

422 Entropy Method Based on the normalized and ben-eficial decisionmatrix Q (qij)mtimesn the attribute weights canbe calculated by means of the entropy method [25]

(a) Calculate the proportions of attributes in matrix QGiven the normalized and beneficial decision matrixQ (qij)mtimesn for attribute Cj the ratio of qij to the sumof all elements in the same column in Q is calculated as

Zij qij

1113936nl1 qlj

i 1 m j 1 n (28)

(b) Calculate the information entropy of the attributes)e information entropy of the attributes can be cal-culated as follows

Ej minusk 1113944m

i1Zij ln zij j 1 n (29)

where k is the adjustment coefficient and k 1ln n(c) Calculate the redundancy of information entropy

ej 1 minus Ej j 1 n (30)

(d) Calculate the weights of attributes

Based on the information entropy of the attributes theirweights can be calculated as follows

Wj ej

1113936nj1 ej

j 1 n (31)

Denote W2 (w21 w2

2 w2n) as the weight vector of

the attributes obtained by means of the entropy method

423 Determine Comprehensive Attribute Weights Basedon the attribute weight vectorW1 calculated by the deviationmaximization model (22a)ndash(22c) and the attribute weightvector W2 obtained by the Entropy method the compre-hensive attribute weight vector based on the decision matrixQ is calculated in the following

Wo

05W1

+ 05W2 (32)

43 Calculate the Overall Values of Suppliers )e overallvalues of supplier Si can be obtained by means of theweighted sum method as follows

overalli 1113944n

j1w

ojqij i 1 m (33)

All the suppliers can be ranked descendingly accordingto their overall values calculated in formula (33)

5 Illustrations

In the course of supplier selection the determination ofassessment attributes is the first step )e selection of at-tributes is reviewed in [9] In this study service level (C1)degree of informatization (C2) profitability (C3) level ofquality (C4) and level of risk (C5) are adopted in evaluatingand selecting four suppliers (ie Si i 1 2 3 and 4) in B2Be-commerce environment Because of the uncertainty andfuzziness in B2B e-commerce environment for the attri-butes of service level (C1) and level of risk (C5) preferenceorderings are used

For the attributes of degree of informatization (C2) andlevel of quality (C4) linguistic terms are used to assess thesupplier performances For the attribute of profitability C3interval numbers are employed For the sake of simplicitythe linguistic term set ldquovery poorrdquo ldquopoorrdquo ldquofairrdquo ldquogoodrdquoand ldquovery goodrdquo is employed for both attributes of degree ofinformatization (C2) and level of quality (C4) and is same asthe basic linguistic term set TERMSETB in this study Detailsof the assessment information of the suppliers (Si i 1 2 34) against the attributes are stated in Table 1

Firstly with respect to attribute C1 of the service levelthe assessment information of the suppliers in Table 1 isnormalized as (b11 b21 b31 b41)T (1 0 03333 06667)Tandis further transformed as (q11 q21 q31 q41)T (1 0333304286 06)T by calculating the grey relational coefficientsbetween their crisp values bi1 (i 1 2 3 4) and the corre-sponding positive ideal attribute value

Secondly with respect to attribute C2 of degree of infor-matization the linguistic assessment information of the suppliersin Table 1 is normalized as (b12 b22 b32 b42)T (09167 025 05075)T and is further transformed as (q12 q22 q32 q42)T (1 0405714 1)T

)irdly with respect to attribute C3 of profitability theinterval assessment information of the suppliers in Table 1 isnormalized as

bl13 bu

13

bl23 bu

23

bl33 bu

33

bl43 bu

43

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

075 1

0 025

05 075

025 05

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

and it is further transformed as (q13 q23 q33 q43)T (0583605182 1 06940)T

Fourthly with respect to attribute C4 of level ofquality the linguistic assessment information of thesuppliers in Table 1 is normalized as (b14 b24 b34 b44)T (05 00833 075 09167)T and is further transformedas (q14 q24 q34 q44)T (05 03333 07143 1)T Fifthlywith respect to attribute C5 of level of risk the assessmentinformation of the suppliers in Table 1 is normalized into

6 Complexity

(b15 b25 b35 b45)T (06667 1 0 03333)T and is furthertransformed as (q15 q25 q35 q45)T (06 1 0333304286)T

Based on the above calculation the single-point valuedecision matrix Q is obtained as

Q

10000 10000 05836 05000 06000

03333 04000 05182 03333 10000

04286 05714 10000 07143 03333

06000 10000 06940 10000 04286

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (35)

Furthermore based on the normalized decision matrixQthe attribute weight vector can be calculated by the deviationmaximization model (22a)ndash(22c) as W1 (02100 0215501504 02141 and 02100) In the meantime the attributeweight vector can also be obtained by the entropy weightmethod as W2 (02120 01970 01746 02044 and 02120))us the comprehensive attribute weight vector based on thedecision matrix Q is calculated as Wo (02110 0206201625 02092 and 02110) Accordingly the overall values ofthe suppliers can be obtained as follows overall1 07432overall2 05177 overall3 06295 and overall4 07452 Fi-nally the ranking of the suppliers is S4gt S1gt S3gt S2

6 Conclusions

)is paper proposes an approach to evaluating and selectingthe manufacturing suppliers in B2B e-commerce environ-ment where linguistic terms preference orderings andinterval numbers are employed to present their fuzzy per-formances After the hybrid decision matrix is normalizedthe attribute weights are determined by means of proposingthe deviation maximization model and the entropy method

)e merits of the proposed approach lie in three aspects)e first one is to express the uncertainty of the suppliersrsquoperformances by means of the appropriate and the easiestways ie linguistic terms preference orderings and intervalnumbers )e second one is to propose the methods ofnormalizing the hybrid decision matrix by calculating thegrey correlation coefficients of attribute values with the idealvalues of attributes )e third one is to determine the at-tribute weights by means of the deviation maximizationmodel and the entropy method based on normalized de-cision matrix )is paper enables to express the suppliersrsquoperformance information in the easiest ways and accuratelyespecially in fuzzy or uncertain decision environmentCompared with the current research the proposed approachhas more universal significance and practical applicationprospects

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is work was supported by the Science and TechnologyAgency of Liaoning Province under the Grant of the NaturalScience Foundation of Liaoning Province (2013020022)ldquoHybrid multicriteria group decision-making with variousforms of information expressionrdquo

References

[1] D Maffin and P Braiden ldquoManufacturing and supplier rolesin product developmentrdquo International Journal of ProductionEconomics vol 69 no 2 pp 205ndash213 2001

[2] G Svensson T Mysen and J Payan ldquoBalancing the se-quential logic of quality constructs in manufacturing-supplierrelationshipsmdashcauses and outcomesrdquo Journal of BusinessResearch vol 63 no 11 pp 1209ndash1214 2010

[3] M J Saenz D Knoppen and E M Tachizawa ldquoBuildingmanufacturing flexibility with strategic suppliers and con-tingent effect of product dynamism on customer satisfactionrdquoJournal of Purchasing and Supply Management vol 24 no 3pp 238ndash246 2018

[4] T-C Chu and R Varma ldquoEvaluating suppliers via a multiplelevels multiple criteria decision making method under fuzzyenvironmentrdquo Computers amp Industrial Engineering vol 62no 2 pp 653ndash660 2012

[5] L Xie J Ma and H Han ldquoImplications of stochastic demandand manufacturersrsquo operational mode on retailerrsquos mixedbundling strategy and its complexity analysisrdquo AppliedMathematical Modelling vol 55 no 1 pp 484ndash501 2018

[6] J Ma and L Xie ldquo)e stability analysis of the dynamic pricingstrategy for bundling goods a comparison between simul-taneous and sequential pricing mechanismrdquo Nonlinear Dy-namics vol 95 no 2 pp 1147ndash1164 2019

[7] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computational Linguistics vol 9 no 1pp 149ndash184 1983

[8] L F D O M Santos L Osiro and R H P Lima ldquoA modelbased on 2-tuple fuzzy linguistic representation and AnalyticHierarchy Process for supplier segmentation using qualitativeand quantitative criteriardquo Expert Systems with Applicationsvol 79 pp 53ndash64 2017

[9] P Sureeyatanapas K Sriwattananusart T NiyamosothW Sessomboon and S Arunyanart ldquoSupplier selection to-wards uncertain and unavailable information an extension of

Table 1 )e assessment information of the suppliers

Service level C1 Degree of informatization C2 Profitability C3 Level of quality C4 Level of risk C5

S1 1 Poor [70 80] Fair 2S2 4 Very poor [40 50] Poor 1S3 3 Fair [60 70] Very good 4S4 2 Very good [50 60] Good 3

Complexity 7

TOPSIS methodrdquo Operations Research Perspectives vol 5pp 69ndash79 2018

[10] C-N Liao and H-P Kao ldquoAn integrated fuzzy TOPSIS andMCGP approach to supplier selection in supply chainmanagementrdquo Expert Systems with Applications vol 38 no 9pp 10803ndash10811 2011

[11] F R L Junior L Osiro and L C R Carpinetti ldquoA com-parison between fuzzy AHP and fuzzy TOPSIS methods tosupplier selectionrdquo Applied Soft Computing vol 21 no 5pp 194ndash209 2014

[12] A Azizi D O Aikhuele and F S Souleman ldquoA fuzzy TOPSISmodel to rank automotive suppliersrdquo Procedia Manufactur-ing vol 2 pp 159ndash164 2015

[13] A Kokangul and Z Susuz ldquoIntegrated analytical hierarchprocess and mathematical programming to supplier selectionproblem with quantity discountrdquo Applied MathematicalModelling vol 33 no 3 pp 1417ndash1429 2009

[14] N R Ware S P Singh and D K Banwet ldquoA mixed-integernon-linear program to model dynamic supplier selectionproblemrdquo Expert Systems with Applications vol 41 no 2pp 671ndash678 2014

[15] H Mirzaee B Naderi and S H R Pasandideh ldquoA pre-emptive fuzzy goal programming model for generalizedsupplier selection and order allocation with incrementaldiscountrdquo Computers amp Industrial Engineering vol 122pp 292ndash302 2018

[16] C L Hwang and K Yoon Multiple Attribute DecisionMaking Methods and Applications Springer-Verlag NewYork NY USA 1981

[17] C C Lee and C Ou-Yang ldquoA neural networks approach forforecasting the supplierrsquos bid prices in supplier selectionnegotiation processrdquo Expert Systems with Applicationsvol 36 no 2 pp 2961ndash2970 2009

[18] A F Guneri T Ertay and A Yucel ldquoAn approach based onANFIS input selection and modeling for supplier selectionproblemrdquo Expert Systems with Applications vol 38 no 12pp 14907ndash14917 2011

[19] C Bai and J Sarkis ldquoIntegrating sustainability into supplierselection with grey system and rough set methodologiesrdquoInternational Journal of Production Economics vol 124 no 1pp 252ndash264 2010

[20] D Golmohammadi and M Mellat-Parast ldquoDeveloping agrey-based decision-making model for supplier selectionrdquoInternational Journal of Production Economics vol 137 no 2pp 191ndash200 2012

[21] M S Memon Y H Lee and S I Mari ldquoGroup multi-criteriasupplier selection using combined grey systems theory anduncertainty theoryrdquo Expert Systems with Applications vol 42no 21 pp 7951ndash7959 2015

[22] B S Ahn and K S Park ldquoComparing methods for multi-attribute decisionmaking with ordinal weightsrdquo Computers ampOperations Research vol 35 no 5 pp 1660ndash1670 2008

[23] J Wang and S Zionts ldquoUsing ordinal data to estimate car-dinal valuesrdquo Journal of Multi-Criteria Decision Analysisvol 22 no 3ndash4 pp 185ndash196 2015

[24] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Eu-ropean Journal of Operational Research vol 166 no 1pp 115ndash132 2005

[25] Y Ji G H Huang and W Sun ldquoRisk assessment of hy-dropower stations through an integrated fuzzy entropy-weight multiple criteria decision making method a case studyof the Xiangxi Riverrdquo Expert Systems with Applicationsvol 42 no 12 pp 5380ndash5389 2015

8 Complexity

[bLij bU

ij] and the corresponding positive ideal attribute valueideallowastj are defined as

qij miniminj dis3 1113957bij ideal

lowastj1113872 11138731113966 1113967 + ρ maximaxj dis3 1113957bij ideal

lowastj1113872 11138731113966 1113967

dis3 1113957bij ideallowastj1113872 1113873 + ρ maximaxj dis3 1113957bij ideal

lowastj1113872 11138731113966 1113967

i 1 m j isin J3 (18)

where dis3(middot) is the distance function between two intervalnumbers as defined in (15) and ρ is the parameter whichusually has a value of 05

42 Determine Attribute Weights After the hybrid decisionmatrix 1113957A [1113957aij]mtimesn is transformed into Q [qij]mtimesn in thissection a deviation maximization model is proposed firstlyto determine the attribute weights and integrate them withthose derived by the entropy method

421 lte Proposed Deviation Maximization Model

Definition 5 Given the normalized and beneficial decisionmatrix Q (qij)mtimesn for attribute Cj the weighted distancebetween alternative Si and all other alternatives is defined as

Dij(W) 1113944m

k1dev qij qkj1113872 1113873wj i 1 m j 1 n

(19)

where W (w1 w2 wn) is the weight vector of the at-tributes and dev (middot) is the difference function between twoattribute values and is defined as follows

dev qij qkj1113872 1113873 qij minus qkj

11138681113868111386811138681113868

11138681113868111386811138681113868 i k 1 m j 1 n

(20)

Definition 6 Given the normalized and beneficial decisionmatrix Q (qij)mtimesn for attribute Cj the weighted distancebetween all alternatives and others is defined as

Dj(W) 1113944m

i1Dij(W) 1113944

m

i11113944

m

k1dev qij qkj1113872 1113873wj j 1 n

(21)

It can be seen that Dj(w) denotes the weighted distancesamong all alternatives for attribute Cj j 1 n As-suming that all the alternatives are equally competitive andthere is no preference between them the optimal weightvector W(W (w1 w2 wn)) of the attributes shouldmaximize the weighted distances among all alternativesacross all the attributes )erefore the following deviationmaximization model is set up to determine the attributeweights

maxD(w) 1113944n

j1Dj(w) 1113944

m

i11113944

n

j11113944

m

k1dev qij qkj1113872 1113873wj

(22a)

st

1113944

n

j1w

2j 1 j 1 n (22b)

wj ge 0 j 1 n (22c)

Theorem 1 lte optimal solution to model (22a)ndash(22c) is

wj 1113936

mi1 1113936

mk1 dev qij qkj1113872 1113873

1113936nj1 1113936

mi1 1113936

mk1 dev qij qkj1113872 11138731113872 1113873

21113969 j 1 n

(23)

Proof)e following Lagrange function is constructed

L(W λ) 1113944m

i11113944

n

j11113944

m

k1dev qij qkj1113872 1113873wj +

λ2

1113944

n

j1w

2j minus 1⎞⎠⎛⎝

(24)

where λ is the Lagrange multiplierLet zLzwj 0 and zLzλ 0 the following equation

can be obtained

1113944

m

i11113944

m

k1dev qij qkj1113872 1113873 + λwj 0 j 1 n (25a)

1113944

n

j1w

2j 1 (25b)

By solving the equations composed of (25a) and (25b)the following can be obtained

λ minus 1113944n

j11113944

m

i11113944

n

k1dev qij qkj1113872 1113873⎞⎠⎤⎦

12

⎛⎝⎡⎢⎢⎢⎢⎣ (26a)

wj 1113936

mi1 1113936

mk1 dev qij qkj1113872 1113873

1113936nj1 1113936

mi1 1113936

mk1 dev qij qkj1113872 11138731113872 1113873

21113969 k 1 m

(26b)

Complexity 5

Furthermore normalize the weights given by (26b) andthe weight of attribute Cj(j 1 n) is obtained

w1j

1113936mi1 1113936

mk1 dev qij qkj1113872 1113873

1113936nj1 1113936

mi1 1113936

mk1 dev qij qkj1113872 11138731113872 1113873

j 1 n (27)

)us weight vector W1 (W1 (w11 w1

2 w1n)) of the

attributes can be obtained based on maximizing theweighted distances among all alternatives across theattributes

422 Entropy Method Based on the normalized and ben-eficial decisionmatrix Q (qij)mtimesn the attribute weights canbe calculated by means of the entropy method [25]

(a) Calculate the proportions of attributes in matrix QGiven the normalized and beneficial decision matrixQ (qij)mtimesn for attribute Cj the ratio of qij to the sumof all elements in the same column in Q is calculated as

Zij qij

1113936nl1 qlj

i 1 m j 1 n (28)

(b) Calculate the information entropy of the attributes)e information entropy of the attributes can be cal-culated as follows

Ej minusk 1113944m

i1Zij ln zij j 1 n (29)

where k is the adjustment coefficient and k 1ln n(c) Calculate the redundancy of information entropy

ej 1 minus Ej j 1 n (30)

(d) Calculate the weights of attributes

Based on the information entropy of the attributes theirweights can be calculated as follows

Wj ej

1113936nj1 ej

j 1 n (31)

Denote W2 (w21 w2

2 w2n) as the weight vector of

the attributes obtained by means of the entropy method

423 Determine Comprehensive Attribute Weights Basedon the attribute weight vectorW1 calculated by the deviationmaximization model (22a)ndash(22c) and the attribute weightvector W2 obtained by the Entropy method the compre-hensive attribute weight vector based on the decision matrixQ is calculated in the following

Wo

05W1

+ 05W2 (32)

43 Calculate the Overall Values of Suppliers )e overallvalues of supplier Si can be obtained by means of theweighted sum method as follows

overalli 1113944n

j1w

ojqij i 1 m (33)

All the suppliers can be ranked descendingly accordingto their overall values calculated in formula (33)

5 Illustrations

In the course of supplier selection the determination ofassessment attributes is the first step )e selection of at-tributes is reviewed in [9] In this study service level (C1)degree of informatization (C2) profitability (C3) level ofquality (C4) and level of risk (C5) are adopted in evaluatingand selecting four suppliers (ie Si i 1 2 3 and 4) in B2Be-commerce environment Because of the uncertainty andfuzziness in B2B e-commerce environment for the attri-butes of service level (C1) and level of risk (C5) preferenceorderings are used

For the attributes of degree of informatization (C2) andlevel of quality (C4) linguistic terms are used to assess thesupplier performances For the attribute of profitability C3interval numbers are employed For the sake of simplicitythe linguistic term set ldquovery poorrdquo ldquopoorrdquo ldquofairrdquo ldquogoodrdquoand ldquovery goodrdquo is employed for both attributes of degree ofinformatization (C2) and level of quality (C4) and is same asthe basic linguistic term set TERMSETB in this study Detailsof the assessment information of the suppliers (Si i 1 2 34) against the attributes are stated in Table 1

Firstly with respect to attribute C1 of the service levelthe assessment information of the suppliers in Table 1 isnormalized as (b11 b21 b31 b41)T (1 0 03333 06667)Tandis further transformed as (q11 q21 q31 q41)T (1 0333304286 06)T by calculating the grey relational coefficientsbetween their crisp values bi1 (i 1 2 3 4) and the corre-sponding positive ideal attribute value

Secondly with respect to attribute C2 of degree of infor-matization the linguistic assessment information of the suppliersin Table 1 is normalized as (b12 b22 b32 b42)T (09167 025 05075)T and is further transformed as (q12 q22 q32 q42)T (1 0405714 1)T

)irdly with respect to attribute C3 of profitability theinterval assessment information of the suppliers in Table 1 isnormalized as

bl13 bu

13

bl23 bu

23

bl33 bu

33

bl43 bu

43

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

075 1

0 025

05 075

025 05

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

and it is further transformed as (q13 q23 q33 q43)T (0583605182 1 06940)T

Fourthly with respect to attribute C4 of level ofquality the linguistic assessment information of thesuppliers in Table 1 is normalized as (b14 b24 b34 b44)T (05 00833 075 09167)T and is further transformedas (q14 q24 q34 q44)T (05 03333 07143 1)T Fifthlywith respect to attribute C5 of level of risk the assessmentinformation of the suppliers in Table 1 is normalized into

6 Complexity

(b15 b25 b35 b45)T (06667 1 0 03333)T and is furthertransformed as (q15 q25 q35 q45)T (06 1 0333304286)T

Based on the above calculation the single-point valuedecision matrix Q is obtained as

Q

10000 10000 05836 05000 06000

03333 04000 05182 03333 10000

04286 05714 10000 07143 03333

06000 10000 06940 10000 04286

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (35)

Furthermore based on the normalized decision matrixQthe attribute weight vector can be calculated by the deviationmaximization model (22a)ndash(22c) as W1 (02100 0215501504 02141 and 02100) In the meantime the attributeweight vector can also be obtained by the entropy weightmethod as W2 (02120 01970 01746 02044 and 02120))us the comprehensive attribute weight vector based on thedecision matrix Q is calculated as Wo (02110 0206201625 02092 and 02110) Accordingly the overall values ofthe suppliers can be obtained as follows overall1 07432overall2 05177 overall3 06295 and overall4 07452 Fi-nally the ranking of the suppliers is S4gt S1gt S3gt S2

6 Conclusions

)is paper proposes an approach to evaluating and selectingthe manufacturing suppliers in B2B e-commerce environ-ment where linguistic terms preference orderings andinterval numbers are employed to present their fuzzy per-formances After the hybrid decision matrix is normalizedthe attribute weights are determined by means of proposingthe deviation maximization model and the entropy method

)e merits of the proposed approach lie in three aspects)e first one is to express the uncertainty of the suppliersrsquoperformances by means of the appropriate and the easiestways ie linguistic terms preference orderings and intervalnumbers )e second one is to propose the methods ofnormalizing the hybrid decision matrix by calculating thegrey correlation coefficients of attribute values with the idealvalues of attributes )e third one is to determine the at-tribute weights by means of the deviation maximizationmodel and the entropy method based on normalized de-cision matrix )is paper enables to express the suppliersrsquoperformance information in the easiest ways and accuratelyespecially in fuzzy or uncertain decision environmentCompared with the current research the proposed approachhas more universal significance and practical applicationprospects

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is work was supported by the Science and TechnologyAgency of Liaoning Province under the Grant of the NaturalScience Foundation of Liaoning Province (2013020022)ldquoHybrid multicriteria group decision-making with variousforms of information expressionrdquo

References

[1] D Maffin and P Braiden ldquoManufacturing and supplier rolesin product developmentrdquo International Journal of ProductionEconomics vol 69 no 2 pp 205ndash213 2001

[2] G Svensson T Mysen and J Payan ldquoBalancing the se-quential logic of quality constructs in manufacturing-supplierrelationshipsmdashcauses and outcomesrdquo Journal of BusinessResearch vol 63 no 11 pp 1209ndash1214 2010

[3] M J Saenz D Knoppen and E M Tachizawa ldquoBuildingmanufacturing flexibility with strategic suppliers and con-tingent effect of product dynamism on customer satisfactionrdquoJournal of Purchasing and Supply Management vol 24 no 3pp 238ndash246 2018

[4] T-C Chu and R Varma ldquoEvaluating suppliers via a multiplelevels multiple criteria decision making method under fuzzyenvironmentrdquo Computers amp Industrial Engineering vol 62no 2 pp 653ndash660 2012

[5] L Xie J Ma and H Han ldquoImplications of stochastic demandand manufacturersrsquo operational mode on retailerrsquos mixedbundling strategy and its complexity analysisrdquo AppliedMathematical Modelling vol 55 no 1 pp 484ndash501 2018

[6] J Ma and L Xie ldquo)e stability analysis of the dynamic pricingstrategy for bundling goods a comparison between simul-taneous and sequential pricing mechanismrdquo Nonlinear Dy-namics vol 95 no 2 pp 1147ndash1164 2019

[7] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computational Linguistics vol 9 no 1pp 149ndash184 1983

[8] L F D O M Santos L Osiro and R H P Lima ldquoA modelbased on 2-tuple fuzzy linguistic representation and AnalyticHierarchy Process for supplier segmentation using qualitativeand quantitative criteriardquo Expert Systems with Applicationsvol 79 pp 53ndash64 2017

[9] P Sureeyatanapas K Sriwattananusart T NiyamosothW Sessomboon and S Arunyanart ldquoSupplier selection to-wards uncertain and unavailable information an extension of

Table 1 )e assessment information of the suppliers

Service level C1 Degree of informatization C2 Profitability C3 Level of quality C4 Level of risk C5

S1 1 Poor [70 80] Fair 2S2 4 Very poor [40 50] Poor 1S3 3 Fair [60 70] Very good 4S4 2 Very good [50 60] Good 3

Complexity 7

TOPSIS methodrdquo Operations Research Perspectives vol 5pp 69ndash79 2018

[10] C-N Liao and H-P Kao ldquoAn integrated fuzzy TOPSIS andMCGP approach to supplier selection in supply chainmanagementrdquo Expert Systems with Applications vol 38 no 9pp 10803ndash10811 2011

[11] F R L Junior L Osiro and L C R Carpinetti ldquoA com-parison between fuzzy AHP and fuzzy TOPSIS methods tosupplier selectionrdquo Applied Soft Computing vol 21 no 5pp 194ndash209 2014

[12] A Azizi D O Aikhuele and F S Souleman ldquoA fuzzy TOPSISmodel to rank automotive suppliersrdquo Procedia Manufactur-ing vol 2 pp 159ndash164 2015

[13] A Kokangul and Z Susuz ldquoIntegrated analytical hierarchprocess and mathematical programming to supplier selectionproblem with quantity discountrdquo Applied MathematicalModelling vol 33 no 3 pp 1417ndash1429 2009

[14] N R Ware S P Singh and D K Banwet ldquoA mixed-integernon-linear program to model dynamic supplier selectionproblemrdquo Expert Systems with Applications vol 41 no 2pp 671ndash678 2014

[15] H Mirzaee B Naderi and S H R Pasandideh ldquoA pre-emptive fuzzy goal programming model for generalizedsupplier selection and order allocation with incrementaldiscountrdquo Computers amp Industrial Engineering vol 122pp 292ndash302 2018

[16] C L Hwang and K Yoon Multiple Attribute DecisionMaking Methods and Applications Springer-Verlag NewYork NY USA 1981

[17] C C Lee and C Ou-Yang ldquoA neural networks approach forforecasting the supplierrsquos bid prices in supplier selectionnegotiation processrdquo Expert Systems with Applicationsvol 36 no 2 pp 2961ndash2970 2009

[18] A F Guneri T Ertay and A Yucel ldquoAn approach based onANFIS input selection and modeling for supplier selectionproblemrdquo Expert Systems with Applications vol 38 no 12pp 14907ndash14917 2011

[19] C Bai and J Sarkis ldquoIntegrating sustainability into supplierselection with grey system and rough set methodologiesrdquoInternational Journal of Production Economics vol 124 no 1pp 252ndash264 2010

[20] D Golmohammadi and M Mellat-Parast ldquoDeveloping agrey-based decision-making model for supplier selectionrdquoInternational Journal of Production Economics vol 137 no 2pp 191ndash200 2012

[21] M S Memon Y H Lee and S I Mari ldquoGroup multi-criteriasupplier selection using combined grey systems theory anduncertainty theoryrdquo Expert Systems with Applications vol 42no 21 pp 7951ndash7959 2015

[22] B S Ahn and K S Park ldquoComparing methods for multi-attribute decisionmaking with ordinal weightsrdquo Computers ampOperations Research vol 35 no 5 pp 1660ndash1670 2008

[23] J Wang and S Zionts ldquoUsing ordinal data to estimate car-dinal valuesrdquo Journal of Multi-Criteria Decision Analysisvol 22 no 3ndash4 pp 185ndash196 2015

[24] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Eu-ropean Journal of Operational Research vol 166 no 1pp 115ndash132 2005

[25] Y Ji G H Huang and W Sun ldquoRisk assessment of hy-dropower stations through an integrated fuzzy entropy-weight multiple criteria decision making method a case studyof the Xiangxi Riverrdquo Expert Systems with Applicationsvol 42 no 12 pp 5380ndash5389 2015

8 Complexity

Furthermore normalize the weights given by (26b) andthe weight of attribute Cj(j 1 n) is obtained

w1j

1113936mi1 1113936

mk1 dev qij qkj1113872 1113873

1113936nj1 1113936

mi1 1113936

mk1 dev qij qkj1113872 11138731113872 1113873

j 1 n (27)

)us weight vector W1 (W1 (w11 w1

2 w1n)) of the

attributes can be obtained based on maximizing theweighted distances among all alternatives across theattributes

422 Entropy Method Based on the normalized and ben-eficial decisionmatrix Q (qij)mtimesn the attribute weights canbe calculated by means of the entropy method [25]

(a) Calculate the proportions of attributes in matrix QGiven the normalized and beneficial decision matrixQ (qij)mtimesn for attribute Cj the ratio of qij to the sumof all elements in the same column in Q is calculated as

Zij qij

1113936nl1 qlj

i 1 m j 1 n (28)

(b) Calculate the information entropy of the attributes)e information entropy of the attributes can be cal-culated as follows

Ej minusk 1113944m

i1Zij ln zij j 1 n (29)

where k is the adjustment coefficient and k 1ln n(c) Calculate the redundancy of information entropy

ej 1 minus Ej j 1 n (30)

(d) Calculate the weights of attributes

Based on the information entropy of the attributes theirweights can be calculated as follows

Wj ej

1113936nj1 ej

j 1 n (31)

Denote W2 (w21 w2

2 w2n) as the weight vector of

the attributes obtained by means of the entropy method

423 Determine Comprehensive Attribute Weights Basedon the attribute weight vectorW1 calculated by the deviationmaximization model (22a)ndash(22c) and the attribute weightvector W2 obtained by the Entropy method the compre-hensive attribute weight vector based on the decision matrixQ is calculated in the following

Wo

05W1

+ 05W2 (32)

43 Calculate the Overall Values of Suppliers )e overallvalues of supplier Si can be obtained by means of theweighted sum method as follows

overalli 1113944n

j1w

ojqij i 1 m (33)

All the suppliers can be ranked descendingly accordingto their overall values calculated in formula (33)

5 Illustrations

In the course of supplier selection the determination ofassessment attributes is the first step )e selection of at-tributes is reviewed in [9] In this study service level (C1)degree of informatization (C2) profitability (C3) level ofquality (C4) and level of risk (C5) are adopted in evaluatingand selecting four suppliers (ie Si i 1 2 3 and 4) in B2Be-commerce environment Because of the uncertainty andfuzziness in B2B e-commerce environment for the attri-butes of service level (C1) and level of risk (C5) preferenceorderings are used

For the attributes of degree of informatization (C2) andlevel of quality (C4) linguistic terms are used to assess thesupplier performances For the attribute of profitability C3interval numbers are employed For the sake of simplicitythe linguistic term set ldquovery poorrdquo ldquopoorrdquo ldquofairrdquo ldquogoodrdquoand ldquovery goodrdquo is employed for both attributes of degree ofinformatization (C2) and level of quality (C4) and is same asthe basic linguistic term set TERMSETB in this study Detailsof the assessment information of the suppliers (Si i 1 2 34) against the attributes are stated in Table 1

Firstly with respect to attribute C1 of the service levelthe assessment information of the suppliers in Table 1 isnormalized as (b11 b21 b31 b41)T (1 0 03333 06667)Tandis further transformed as (q11 q21 q31 q41)T (1 0333304286 06)T by calculating the grey relational coefficientsbetween their crisp values bi1 (i 1 2 3 4) and the corre-sponding positive ideal attribute value

Secondly with respect to attribute C2 of degree of infor-matization the linguistic assessment information of the suppliersin Table 1 is normalized as (b12 b22 b32 b42)T (09167 025 05075)T and is further transformed as (q12 q22 q32 q42)T (1 0405714 1)T

)irdly with respect to attribute C3 of profitability theinterval assessment information of the suppliers in Table 1 isnormalized as

bl13 bu

13

bl23 bu

23

bl33 bu

33

bl43 bu

43

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

075 1

0 025

05 075

025 05

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

and it is further transformed as (q13 q23 q33 q43)T (0583605182 1 06940)T

Fourthly with respect to attribute C4 of level ofquality the linguistic assessment information of thesuppliers in Table 1 is normalized as (b14 b24 b34 b44)T (05 00833 075 09167)T and is further transformedas (q14 q24 q34 q44)T (05 03333 07143 1)T Fifthlywith respect to attribute C5 of level of risk the assessmentinformation of the suppliers in Table 1 is normalized into

6 Complexity

(b15 b25 b35 b45)T (06667 1 0 03333)T and is furthertransformed as (q15 q25 q35 q45)T (06 1 0333304286)T

Based on the above calculation the single-point valuedecision matrix Q is obtained as

Q

10000 10000 05836 05000 06000

03333 04000 05182 03333 10000

04286 05714 10000 07143 03333

06000 10000 06940 10000 04286

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (35)

Furthermore based on the normalized decision matrixQthe attribute weight vector can be calculated by the deviationmaximization model (22a)ndash(22c) as W1 (02100 0215501504 02141 and 02100) In the meantime the attributeweight vector can also be obtained by the entropy weightmethod as W2 (02120 01970 01746 02044 and 02120))us the comprehensive attribute weight vector based on thedecision matrix Q is calculated as Wo (02110 0206201625 02092 and 02110) Accordingly the overall values ofthe suppliers can be obtained as follows overall1 07432overall2 05177 overall3 06295 and overall4 07452 Fi-nally the ranking of the suppliers is S4gt S1gt S3gt S2

6 Conclusions

)is paper proposes an approach to evaluating and selectingthe manufacturing suppliers in B2B e-commerce environ-ment where linguistic terms preference orderings andinterval numbers are employed to present their fuzzy per-formances After the hybrid decision matrix is normalizedthe attribute weights are determined by means of proposingthe deviation maximization model and the entropy method

)e merits of the proposed approach lie in three aspects)e first one is to express the uncertainty of the suppliersrsquoperformances by means of the appropriate and the easiestways ie linguistic terms preference orderings and intervalnumbers )e second one is to propose the methods ofnormalizing the hybrid decision matrix by calculating thegrey correlation coefficients of attribute values with the idealvalues of attributes )e third one is to determine the at-tribute weights by means of the deviation maximizationmodel and the entropy method based on normalized de-cision matrix )is paper enables to express the suppliersrsquoperformance information in the easiest ways and accuratelyespecially in fuzzy or uncertain decision environmentCompared with the current research the proposed approachhas more universal significance and practical applicationprospects

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is work was supported by the Science and TechnologyAgency of Liaoning Province under the Grant of the NaturalScience Foundation of Liaoning Province (2013020022)ldquoHybrid multicriteria group decision-making with variousforms of information expressionrdquo

References

[1] D Maffin and P Braiden ldquoManufacturing and supplier rolesin product developmentrdquo International Journal of ProductionEconomics vol 69 no 2 pp 205ndash213 2001

[2] G Svensson T Mysen and J Payan ldquoBalancing the se-quential logic of quality constructs in manufacturing-supplierrelationshipsmdashcauses and outcomesrdquo Journal of BusinessResearch vol 63 no 11 pp 1209ndash1214 2010

[3] M J Saenz D Knoppen and E M Tachizawa ldquoBuildingmanufacturing flexibility with strategic suppliers and con-tingent effect of product dynamism on customer satisfactionrdquoJournal of Purchasing and Supply Management vol 24 no 3pp 238ndash246 2018

[4] T-C Chu and R Varma ldquoEvaluating suppliers via a multiplelevels multiple criteria decision making method under fuzzyenvironmentrdquo Computers amp Industrial Engineering vol 62no 2 pp 653ndash660 2012

[5] L Xie J Ma and H Han ldquoImplications of stochastic demandand manufacturersrsquo operational mode on retailerrsquos mixedbundling strategy and its complexity analysisrdquo AppliedMathematical Modelling vol 55 no 1 pp 484ndash501 2018

[6] J Ma and L Xie ldquo)e stability analysis of the dynamic pricingstrategy for bundling goods a comparison between simul-taneous and sequential pricing mechanismrdquo Nonlinear Dy-namics vol 95 no 2 pp 1147ndash1164 2019

[7] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computational Linguistics vol 9 no 1pp 149ndash184 1983

[8] L F D O M Santos L Osiro and R H P Lima ldquoA modelbased on 2-tuple fuzzy linguistic representation and AnalyticHierarchy Process for supplier segmentation using qualitativeand quantitative criteriardquo Expert Systems with Applicationsvol 79 pp 53ndash64 2017

[9] P Sureeyatanapas K Sriwattananusart T NiyamosothW Sessomboon and S Arunyanart ldquoSupplier selection to-wards uncertain and unavailable information an extension of

Table 1 )e assessment information of the suppliers

Service level C1 Degree of informatization C2 Profitability C3 Level of quality C4 Level of risk C5

S1 1 Poor [70 80] Fair 2S2 4 Very poor [40 50] Poor 1S3 3 Fair [60 70] Very good 4S4 2 Very good [50 60] Good 3

Complexity 7

TOPSIS methodrdquo Operations Research Perspectives vol 5pp 69ndash79 2018

[10] C-N Liao and H-P Kao ldquoAn integrated fuzzy TOPSIS andMCGP approach to supplier selection in supply chainmanagementrdquo Expert Systems with Applications vol 38 no 9pp 10803ndash10811 2011

[11] F R L Junior L Osiro and L C R Carpinetti ldquoA com-parison between fuzzy AHP and fuzzy TOPSIS methods tosupplier selectionrdquo Applied Soft Computing vol 21 no 5pp 194ndash209 2014

[12] A Azizi D O Aikhuele and F S Souleman ldquoA fuzzy TOPSISmodel to rank automotive suppliersrdquo Procedia Manufactur-ing vol 2 pp 159ndash164 2015

[13] A Kokangul and Z Susuz ldquoIntegrated analytical hierarchprocess and mathematical programming to supplier selectionproblem with quantity discountrdquo Applied MathematicalModelling vol 33 no 3 pp 1417ndash1429 2009

[14] N R Ware S P Singh and D K Banwet ldquoA mixed-integernon-linear program to model dynamic supplier selectionproblemrdquo Expert Systems with Applications vol 41 no 2pp 671ndash678 2014

[15] H Mirzaee B Naderi and S H R Pasandideh ldquoA pre-emptive fuzzy goal programming model for generalizedsupplier selection and order allocation with incrementaldiscountrdquo Computers amp Industrial Engineering vol 122pp 292ndash302 2018

[16] C L Hwang and K Yoon Multiple Attribute DecisionMaking Methods and Applications Springer-Verlag NewYork NY USA 1981

[17] C C Lee and C Ou-Yang ldquoA neural networks approach forforecasting the supplierrsquos bid prices in supplier selectionnegotiation processrdquo Expert Systems with Applicationsvol 36 no 2 pp 2961ndash2970 2009

[18] A F Guneri T Ertay and A Yucel ldquoAn approach based onANFIS input selection and modeling for supplier selectionproblemrdquo Expert Systems with Applications vol 38 no 12pp 14907ndash14917 2011

[19] C Bai and J Sarkis ldquoIntegrating sustainability into supplierselection with grey system and rough set methodologiesrdquoInternational Journal of Production Economics vol 124 no 1pp 252ndash264 2010

[20] D Golmohammadi and M Mellat-Parast ldquoDeveloping agrey-based decision-making model for supplier selectionrdquoInternational Journal of Production Economics vol 137 no 2pp 191ndash200 2012

[21] M S Memon Y H Lee and S I Mari ldquoGroup multi-criteriasupplier selection using combined grey systems theory anduncertainty theoryrdquo Expert Systems with Applications vol 42no 21 pp 7951ndash7959 2015

[22] B S Ahn and K S Park ldquoComparing methods for multi-attribute decisionmaking with ordinal weightsrdquo Computers ampOperations Research vol 35 no 5 pp 1660ndash1670 2008

[23] J Wang and S Zionts ldquoUsing ordinal data to estimate car-dinal valuesrdquo Journal of Multi-Criteria Decision Analysisvol 22 no 3ndash4 pp 185ndash196 2015

[24] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Eu-ropean Journal of Operational Research vol 166 no 1pp 115ndash132 2005

[25] Y Ji G H Huang and W Sun ldquoRisk assessment of hy-dropower stations through an integrated fuzzy entropy-weight multiple criteria decision making method a case studyof the Xiangxi Riverrdquo Expert Systems with Applicationsvol 42 no 12 pp 5380ndash5389 2015

8 Complexity

(b15 b25 b35 b45)T (06667 1 0 03333)T and is furthertransformed as (q15 q25 q35 q45)T (06 1 0333304286)T

Based on the above calculation the single-point valuedecision matrix Q is obtained as

Q

10000 10000 05836 05000 06000

03333 04000 05182 03333 10000

04286 05714 10000 07143 03333

06000 10000 06940 10000 04286

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (35)

Furthermore based on the normalized decision matrixQthe attribute weight vector can be calculated by the deviationmaximization model (22a)ndash(22c) as W1 (02100 0215501504 02141 and 02100) In the meantime the attributeweight vector can also be obtained by the entropy weightmethod as W2 (02120 01970 01746 02044 and 02120))us the comprehensive attribute weight vector based on thedecision matrix Q is calculated as Wo (02110 0206201625 02092 and 02110) Accordingly the overall values ofthe suppliers can be obtained as follows overall1 07432overall2 05177 overall3 06295 and overall4 07452 Fi-nally the ranking of the suppliers is S4gt S1gt S3gt S2

6 Conclusions

)is paper proposes an approach to evaluating and selectingthe manufacturing suppliers in B2B e-commerce environ-ment where linguistic terms preference orderings andinterval numbers are employed to present their fuzzy per-formances After the hybrid decision matrix is normalizedthe attribute weights are determined by means of proposingthe deviation maximization model and the entropy method

)e merits of the proposed approach lie in three aspects)e first one is to express the uncertainty of the suppliersrsquoperformances by means of the appropriate and the easiestways ie linguistic terms preference orderings and intervalnumbers )e second one is to propose the methods ofnormalizing the hybrid decision matrix by calculating thegrey correlation coefficients of attribute values with the idealvalues of attributes )e third one is to determine the at-tribute weights by means of the deviation maximizationmodel and the entropy method based on normalized de-cision matrix )is paper enables to express the suppliersrsquoperformance information in the easiest ways and accuratelyespecially in fuzzy or uncertain decision environmentCompared with the current research the proposed approachhas more universal significance and practical applicationprospects

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is work was supported by the Science and TechnologyAgency of Liaoning Province under the Grant of the NaturalScience Foundation of Liaoning Province (2013020022)ldquoHybrid multicriteria group decision-making with variousforms of information expressionrdquo

References

[1] D Maffin and P Braiden ldquoManufacturing and supplier rolesin product developmentrdquo International Journal of ProductionEconomics vol 69 no 2 pp 205ndash213 2001

[2] G Svensson T Mysen and J Payan ldquoBalancing the se-quential logic of quality constructs in manufacturing-supplierrelationshipsmdashcauses and outcomesrdquo Journal of BusinessResearch vol 63 no 11 pp 1209ndash1214 2010

[3] M J Saenz D Knoppen and E M Tachizawa ldquoBuildingmanufacturing flexibility with strategic suppliers and con-tingent effect of product dynamism on customer satisfactionrdquoJournal of Purchasing and Supply Management vol 24 no 3pp 238ndash246 2018

[4] T-C Chu and R Varma ldquoEvaluating suppliers via a multiplelevels multiple criteria decision making method under fuzzyenvironmentrdquo Computers amp Industrial Engineering vol 62no 2 pp 653ndash660 2012

[5] L Xie J Ma and H Han ldquoImplications of stochastic demandand manufacturersrsquo operational mode on retailerrsquos mixedbundling strategy and its complexity analysisrdquo AppliedMathematical Modelling vol 55 no 1 pp 484ndash501 2018

[6] J Ma and L Xie ldquo)e stability analysis of the dynamic pricingstrategy for bundling goods a comparison between simul-taneous and sequential pricing mechanismrdquo Nonlinear Dy-namics vol 95 no 2 pp 1147ndash1164 2019

[7] L A Zadeh ldquoA computational approach to fuzzy quantifiersin natural languagesrdquo Computational Linguistics vol 9 no 1pp 149ndash184 1983

[8] L F D O M Santos L Osiro and R H P Lima ldquoA modelbased on 2-tuple fuzzy linguistic representation and AnalyticHierarchy Process for supplier segmentation using qualitativeand quantitative criteriardquo Expert Systems with Applicationsvol 79 pp 53ndash64 2017

[9] P Sureeyatanapas K Sriwattananusart T NiyamosothW Sessomboon and S Arunyanart ldquoSupplier selection to-wards uncertain and unavailable information an extension of

Table 1 )e assessment information of the suppliers

Service level C1 Degree of informatization C2 Profitability C3 Level of quality C4 Level of risk C5

S1 1 Poor [70 80] Fair 2S2 4 Very poor [40 50] Poor 1S3 3 Fair [60 70] Very good 4S4 2 Very good [50 60] Good 3

Complexity 7

TOPSIS methodrdquo Operations Research Perspectives vol 5pp 69ndash79 2018

[10] C-N Liao and H-P Kao ldquoAn integrated fuzzy TOPSIS andMCGP approach to supplier selection in supply chainmanagementrdquo Expert Systems with Applications vol 38 no 9pp 10803ndash10811 2011

[11] F R L Junior L Osiro and L C R Carpinetti ldquoA com-parison between fuzzy AHP and fuzzy TOPSIS methods tosupplier selectionrdquo Applied Soft Computing vol 21 no 5pp 194ndash209 2014

[12] A Azizi D O Aikhuele and F S Souleman ldquoA fuzzy TOPSISmodel to rank automotive suppliersrdquo Procedia Manufactur-ing vol 2 pp 159ndash164 2015

[13] A Kokangul and Z Susuz ldquoIntegrated analytical hierarchprocess and mathematical programming to supplier selectionproblem with quantity discountrdquo Applied MathematicalModelling vol 33 no 3 pp 1417ndash1429 2009

[14] N R Ware S P Singh and D K Banwet ldquoA mixed-integernon-linear program to model dynamic supplier selectionproblemrdquo Expert Systems with Applications vol 41 no 2pp 671ndash678 2014

[15] H Mirzaee B Naderi and S H R Pasandideh ldquoA pre-emptive fuzzy goal programming model for generalizedsupplier selection and order allocation with incrementaldiscountrdquo Computers amp Industrial Engineering vol 122pp 292ndash302 2018

[16] C L Hwang and K Yoon Multiple Attribute DecisionMaking Methods and Applications Springer-Verlag NewYork NY USA 1981

[17] C C Lee and C Ou-Yang ldquoA neural networks approach forforecasting the supplierrsquos bid prices in supplier selectionnegotiation processrdquo Expert Systems with Applicationsvol 36 no 2 pp 2961ndash2970 2009

[18] A F Guneri T Ertay and A Yucel ldquoAn approach based onANFIS input selection and modeling for supplier selectionproblemrdquo Expert Systems with Applications vol 38 no 12pp 14907ndash14917 2011

[19] C Bai and J Sarkis ldquoIntegrating sustainability into supplierselection with grey system and rough set methodologiesrdquoInternational Journal of Production Economics vol 124 no 1pp 252ndash264 2010

[20] D Golmohammadi and M Mellat-Parast ldquoDeveloping agrey-based decision-making model for supplier selectionrdquoInternational Journal of Production Economics vol 137 no 2pp 191ndash200 2012

[21] M S Memon Y H Lee and S I Mari ldquoGroup multi-criteriasupplier selection using combined grey systems theory anduncertainty theoryrdquo Expert Systems with Applications vol 42no 21 pp 7951ndash7959 2015

[22] B S Ahn and K S Park ldquoComparing methods for multi-attribute decisionmaking with ordinal weightsrdquo Computers ampOperations Research vol 35 no 5 pp 1660ndash1670 2008

[23] J Wang and S Zionts ldquoUsing ordinal data to estimate car-dinal valuesrdquo Journal of Multi-Criteria Decision Analysisvol 22 no 3ndash4 pp 185ndash196 2015

[24] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Eu-ropean Journal of Operational Research vol 166 no 1pp 115ndash132 2005

[25] Y Ji G H Huang and W Sun ldquoRisk assessment of hy-dropower stations through an integrated fuzzy entropy-weight multiple criteria decision making method a case studyof the Xiangxi Riverrdquo Expert Systems with Applicationsvol 42 no 12 pp 5380ndash5389 2015

8 Complexity

TOPSIS methodrdquo Operations Research Perspectives vol 5pp 69ndash79 2018

[10] C-N Liao and H-P Kao ldquoAn integrated fuzzy TOPSIS andMCGP approach to supplier selection in supply chainmanagementrdquo Expert Systems with Applications vol 38 no 9pp 10803ndash10811 2011

[11] F R L Junior L Osiro and L C R Carpinetti ldquoA com-parison between fuzzy AHP and fuzzy TOPSIS methods tosupplier selectionrdquo Applied Soft Computing vol 21 no 5pp 194ndash209 2014

[12] A Azizi D O Aikhuele and F S Souleman ldquoA fuzzy TOPSISmodel to rank automotive suppliersrdquo Procedia Manufactur-ing vol 2 pp 159ndash164 2015

[13] A Kokangul and Z Susuz ldquoIntegrated analytical hierarchprocess and mathematical programming to supplier selectionproblem with quantity discountrdquo Applied MathematicalModelling vol 33 no 3 pp 1417ndash1429 2009

[14] N R Ware S P Singh and D K Banwet ldquoA mixed-integernon-linear program to model dynamic supplier selectionproblemrdquo Expert Systems with Applications vol 41 no 2pp 671ndash678 2014

[15] H Mirzaee B Naderi and S H R Pasandideh ldquoA pre-emptive fuzzy goal programming model for generalizedsupplier selection and order allocation with incrementaldiscountrdquo Computers amp Industrial Engineering vol 122pp 292ndash302 2018

[16] C L Hwang and K Yoon Multiple Attribute DecisionMaking Methods and Applications Springer-Verlag NewYork NY USA 1981

[17] C C Lee and C Ou-Yang ldquoA neural networks approach forforecasting the supplierrsquos bid prices in supplier selectionnegotiation processrdquo Expert Systems with Applicationsvol 36 no 2 pp 2961ndash2970 2009

[18] A F Guneri T Ertay and A Yucel ldquoAn approach based onANFIS input selection and modeling for supplier selectionproblemrdquo Expert Systems with Applications vol 38 no 12pp 14907ndash14917 2011

[19] C Bai and J Sarkis ldquoIntegrating sustainability into supplierselection with grey system and rough set methodologiesrdquoInternational Journal of Production Economics vol 124 no 1pp 252ndash264 2010

[20] D Golmohammadi and M Mellat-Parast ldquoDeveloping agrey-based decision-making model for supplier selectionrdquoInternational Journal of Production Economics vol 137 no 2pp 191ndash200 2012

[21] M S Memon Y H Lee and S I Mari ldquoGroup multi-criteriasupplier selection using combined grey systems theory anduncertainty theoryrdquo Expert Systems with Applications vol 42no 21 pp 7951ndash7959 2015

[22] B S Ahn and K S Park ldquoComparing methods for multi-attribute decisionmaking with ordinal weightsrdquo Computers ampOperations Research vol 35 no 5 pp 1660ndash1670 2008

[23] J Wang and S Zionts ldquoUsing ordinal data to estimate car-dinal valuesrdquo Journal of Multi-Criteria Decision Analysisvol 22 no 3ndash4 pp 185ndash196 2015

[24] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Eu-ropean Journal of Operational Research vol 166 no 1pp 115ndash132 2005

[25] Y Ji G H Huang and W Sun ldquoRisk assessment of hy-dropower stations through an integrated fuzzy entropy-weight multiple criteria decision making method a case studyof the Xiangxi Riverrdquo Expert Systems with Applicationsvol 42 no 12 pp 5380ndash5389 2015

8 Complexity