Research Article Supplier Evaluation Process by Pairwise...

Research ArticleSupplier Evaluation Process by Pairwise Comparisons

Arkadiusz Kawa1 and Waldemar W Koczkodaj2

1Department of Logistics and Transport Poznan University of Economics 61-875 Poznan Poland2Department of Mathematics and Computer Science Laurentian University Sudbury ON Canada P3E 2C6

Correspondence should be addressed to Arkadiusz Kawa arkadiuszkawauepoznanpl

Received 9 June 2015 Accepted 8 December 2015

Academic Editor Ben T Nohara

Copyright copy 2015 A Kawa and W W Koczkodaj This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We propose to assess suppliers by using consistency-driven pairwise comparisons for tangible and intangible criteria The tangiblecriteria are simpler to compare (eg the price of a service is lower than that of another service with identical characteristics)Intangible criteria are more difficult to assess The proposed model combines assessments of both types of criteria The maincontribution of this paper is the presentation of an extension framework for the selection of suppliers in a procurement processThefinal weights are computed from relative pairwise comparisons For the needs of the paper surveys were conducted among Polishmanagers dealing with cooperation with suppliers in their enterprisesThe Polish practice and restricted bidding are discussed too

1 Introduction

The success of an enterprise does not depend only on all thecooperating subjects [1] Efficiency of a given subject as awhole is not its basic source of competitive advantage It isthe efficiency of various types of activities that the enterpriseundertakes when delivering its product to the market [2]These actions create a supply chainThemain objective of thesupply chain is to provide the maximum value to customersat low costs and high speed [3 4]

An important issue of a supply chain management is theprocurement [5] The cost of products and services acquiredfrom external suppliers is significant for most manufacturingfirms [6] On average manufacturers purchases of goods andservices amount to 55 of revenues and it is in contrastto labor costs of 6 and overhead expenses of 3 ofrevenues [7] For high-technology firms purchased materialand services represent up to 80 of total product costs [8 9]

The objective of the procurement process must be theharmonization of internal processes of buyers and suppliersin order to avoid a waste of resources within the supplychain It may be achieved when a great amount of emphasisis put on establishing and maintaining good relations with

suppliers Optimization in the area of delivery may bring achance of sizable savings It is noteworthy that the scale ofa companyrsquos activity may result in greater losses caused byincorrect purchasing processes

Reliable suppliers enable manufacturers to reduce inven-tory level and improve product quality which is themain rea-son why concerns about appropriate suppliers are increasingOne of the prime responsibilities of the purchasing functionis the evaluation and selection of suppliers Some researcherseven indicate that evaluation and selection of suppliers arecritical due to their contribution to supply chain performance[10]

Enterprises build a base of suppliers with whom theycooperate more or less closely for many years When search-ing for solutions related to the choice of a new supplier orevaluation of an existing one people responsible for suchdecisions take into account a range of criteria These criteriaare very often worked out on the basis of the companyrsquosexperience

Carefully selected competitive suppliers can go a longway inminimizing adverse impacts and in fact in enhancingpositive impacts on the quality of the output of an organi-zation The importance of an adequate framework for the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 976742 9 pageshttpdxdoiorg1011552015976742

2 Mathematical Problems in Engineering

selection of suppliers for an organization has been stressedin the literature [11 12]

The supplier selection process requires predeterminationof several issues including the number of suppliers withwhich the organization wishes to contract for a given productor service [13] It is difficult to determine howmany suppliersshould work for a given organization Some authors arguethat there is now a trend to reduce the number of suppliers toa manageable level [14 15] even to a single source in extremecases

No clear recommendation regarding the number of sup-pliers is confirmed by the results of the research Every thirdcompanydoes not follow any general ruleThis is understand-able On the one hand a large number of suppliers ensureslower prices for customers offers greater safety and decreasesthe risk of stopping production On the other hand it raisesthe operating costs of such cooperation (maintenance costsof information systems controlling sourcing negotiationsetting conditions for the cooperation audits etc) [16 17]

In the process of partner selection several methodsare used Among them categorical method weighted-pointmethod vendor performance matrix approach vendor pro-file analysis (VPA) multicriteria decision aid (MCDA) mul-tiple objective programming (MOP) such as goal program-ming data envelopment analysis (DEA) and multiattributeutility theory (MAUT) The recent studies about the useof fuzzy theories and their development in the supplierselection problem are particularly interesting [18ndash20] In [21]the authors proposed a new tool for supplier selection basedon a combination of the grey system and the uncertaintytheory neither of which requires any probability distributionor fuzzy membership function

The structure of the paper is as follows the characteristicof bidding process is discussed in Section 2 Then Sec-tion 3 describes supplier evaluation problems and Section 4presents pairwise comparisons method and review of theother applications and the literature on the pairwise com-parisons Section 5 provides example of a tendering modelproposed by the authors Next Section 6 contains consistencyanalysis of tendering model Section 7 proposes an exampleof suppliers assessment The conclusions of research areoutlined in Section 8

2 Bidding Process

The previously mentioned methods have some limitationsbecause the supplier selection process is mostly based onintuitionThere is no theoretical base or consistentmethod ofpredicting the best bid It is not uncommon for the evaluationpanel to arrive at a deadlock when a part of the panel favorsone solution because of certain criteria while the otherpart insists on another solution since it scores better ondifferent criteria The decision making process nearly alwaysinvolves some kind of constituency in modern democraticsocieties We have various boards of governors or directorscommittees task groups city councils panels of experts andindividuals each with a specific agenda Heated discussionsand various ways of dispute and argumentation often takeplace to arrive at certain decisions

Most constituencies have worked out precise and practi-cal policies for running meetings in an orderly and effectivemanner What we lack however is a device for drawing solidconsistent conclusions and all too often the loudest individualwins Unfortunately loudness does not necessarily go alongwith wisdom Casual thinking is not efficient in predictingcomplex outcomes

The main goal of tendering is the selection of the mostsuitable supplier from the company point of view Through abidding we try to achieve the following

(i) the setting of common input constraints for potentialsuppliers

(ii) the selection of the best supplier based on tangibleand intangible but constant (during the entire biddingprocess) criteria which allow us to compare theproposed offers

(iii) a minimization of the influence of informal interestson selection of an offer thanks to the application of astrict selection process

One of the most instrumental conditions of a fair biddingis the necessity of a precise scoring system of all criteriaand their preferences in the tendering documents to be usedduring the selection process by the selection panel

A selection panel is obligated to use objective and mea-surable evaluation criteria during the selection process Thegoal of such an action is to have more comparability andfewer intangible or arbitrary decisions Application of suchquantification in the request for proposals (RFPs) to specificbidding components together with the bidding amounthelps in the selection of the best offer A recommendedpractice is the assignment of weights (eg factors which arepositive points) to each aspect of an offer with respect to thecriteria listed in the bidding documents As a result the bestoffer is the one with the highest total score

In restricted bidding (as opposed to open bidding) onlyinvited parties may submit a bid In the first stage of atwo-stage bidding process suppliers submit preliminary bidswithout the total amount However the selection panel mayrequest precise specification of some parameters Successfulbidders from the first stage are invited to submit a completebid for the second stage Two-stage bidding may be consid-ered a special case of an open bidding The main differencesare as follows

(i) the possibility of submitting two tendering offers bybidders

(ii) the use of two selections instead of one the firstselection to prequalify tenders and the second to findthe true winner

(iii) the possibility (and sometimes necessity) of negotia-tions with bidders

(iv) the possibility of using the negotiation results tochange or restrain crucial constraints of the requestfor proposals before they are distributed again tobidders selected during the first stage of bidding

Mathematical Problems in Engineering 3

The two-stage bidding process is recommended in situa-tions

(i) where it is difficult to predict certain parameters suchas technical quality service or construction tasks

(ii) where necessity demands negotiating with suppliersor constructors because of the specific character of thesupplies andor construction services

(iii) when the tendering is related to research assessmentor any other specialized service

3 Is There an Easy Way to EvaluatePotential Suppliers

Evaluation problems are not new They have deep rootsin our modern perception of measurement systems Oneof the major achievements of our technical civilization isstandardization It is fair to say that without standardsrapid technological progress would have not taken placeStandardization should not only be perceived as a ldquonuts andboltsrdquo or physical concept (without which new products andspare parts would be far more expensive) but also as anintellectual concept In the case of ldquonuts and boltsrdquo it is aconcept of the standard thread pitch Today we use standardsto such a degree that we often forget about other alternativesThe problem is that we often assume that we have a measurefor nearly everything No one questions the practicality ofthe measure of length (meter or foot) or weight (kg orpound) as they are used in everyday life There are howeversituations where it is more useful and natural to use pairwisecomparisons Consider this example of the reliability of twosuppliers 119860 and 119861 The reliability of supplier 119860 is evaluatedas 4 on the scale of 0 to 5 and 119861 is 2 on the same scaleThere is no standardized unit of reliability hence it is easierto say that 119860 is two times more reliable than 119861 and it iswhat we call ldquopairwise comparisonrdquo Interestingly the use ofa standardized measure unit such as a meter stick is also apairwise comparison The statement ldquothe length of 119860 is 4mrdquois an abbreviation of ldquoby a pairwise comparison of 119860 to onemeter we have a factor of 4rdquo

Using pairwise comparisons (see eg [22ndash24]) is naturaland it does not need to be imprecise We have become soaccustomed to having standards that sometimes we find itdifficult to imagine a situation where no standard measuresexist The truth is that there are many such situations

Measurement of the environment or environmental pol-lution are good examples of situations where a standardyardstick seems to be missing For example it would be hardto use a cubic meter of environment as a standard measuresince in one cubic meter of environment there could bemillions of ants but only a fraction of an elephant How couldone decide if a fraction of an elephant is less significant thana colony of ants

4 Pairwise Comparisons

Casual thinking is partial and fragmentary and is not aneffective way to measure intangibles In the decision making

process many factors must be considered simultaneouslyand with about the same degree of importance As such anapproach with more finesse is necessary to obtain a clear andunambiguous conclusion It has been shown by numerousexamples that the pairwise comparisons method can be usedto draw final conclusions in a comparatively easy and elegantway The brilliance of the pairwise comparisons could bereduced to a rule of common sense consider two objects at atime if you are unable to handle more than that Llull noticedit in 13th century

The practical and theoretical virtue of the pairwise com-parisons methodology is its simplicity The goal of pairwisecomparisons is to establish the relative preference of twocriteria in situations in which it is impractical (or sometimesmeaningless) to provide the absolute estimations of thecriteria To this end an expert (or a team of experts) providesrelative comparison coefficients 119886

119894119895gt 0 which are meant to

be a substitute for the quotients 119904119894119904119895of the unknown (or

even undefined) absolute values of criteria 119904119894 119904119895gt 0 The

quotients 119904119894119904119895are also sometimes called relative weights in

the literature

A =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 11988612sdot sdot sdot 1198861119899

1

11988612

1 sdot sdot sdot 1198862119899

1

1198861119899

1

1198862119899

sdot sdot sdot 1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(1)

where 119886119894119895expresses an expertrsquos relative preference of criteria

119904119894 over 119904

119895

Coefficients 119886119894119895

are expected to satisfy some naturalrestrictions (eg 119886

119894119894= 1 119886

119894119895sdot 119886119895119894= 1) For the sake of our

exposition we define the pairwise comparisons 119899times119899matricessimply as square matricesA = (119886

119894119895) such that 119886

119894119895gt 0 for every

119894 119895 = 1 119899A pairwise comparisons matrix A is called reciprocal if119886119894119895= 1119886

119895119894for every 119894 119895 = 1 119899 (then automatically

119886119894119894= 1 for every 119894 = 1 119899) Even a stronger condition

seems natural A pairwise comparisons matrix A is calledconsistent if 119886

119894119896= 119886119894119895sdot 119886119895119896

for every 119894 119895 119896 = 1 119899 Whileevery consistent matrix is reciprocal the inverse in generalfails Consistent matrices correspond to the ideal situationin which there are exact values 119904

1 119904

119899for criteria The

quotients 119886119894119895= 119904119894119904119895form a consistent matrix Conversely the

starting point of the pairwise comparisons inference theorystates that for every 119899 times 119899 consistent matrix A = (119886

119894119895) there

exists positive real numbers 1199041 119904119899such that 119886

119894119895= 119904119894119904119895for

every 119894 119895 = 1 119899 Such vector 119904 = (1199041 119904119899) is unique up

to a multiplicative constantHow can we establish fair weights Is there a theory to

help us The weighting classification needs to be done on afair basis for every criterion which ought to have its sharein contributing to the overall judgment A fair solution isto compare all criteria in pairs using for example a smallscale from 1 to 3 (as mathematically reasoned on the basisof the Fuloprsquos constant in [25]) presented in Table 1 Thesolution accuracy of not-so-inconsistent matrices depends


Table 1 Comparison scale

Code Definition of intensityor importance Application

1 Equal importance

Two criteria equallycontribute to the objectiveor lack of knowledge tocompare them

2 Essential or strongimportance

Experience and judgmentsfavor one criterion overanother

3 Absolute importanceThe highest affirmationdegree of favoring onecriterion over another

14 and soforth

Intermediatejudgments

When compromise isneeded

on the inconsistency (see [26] for details) Smaller scalecontributes to decreased inconsistencyOnemaynote that theconsistency-driven approach is in brief the next step forwardin the development of pairwise comparisons By concentrat-ing on the consistency as the means to improve the precisionof judgments it picks up where Analytic Hierarchy Processarrived to AHP inconsistency index detected the existenceof the inconsistency by the eigenvalue-based heuristic andits rather questionable threshold of 10 randomly generatedmatrices In [27] a mathematical reasoning (based on twocounter examples) was provided to reject the eigenvalue-based inconsistency

It is not our goal to present the consistency-drivenapproach here but only its application to evaluation ofproposals The theoretical foundations of this method arebased on [28ndash30] while [31] presents convincing statisticalevidence that the pairwise comparisons are contributing tothe improvement of accuracy The pairwise comparisonsmethodologywas introduced byThurstone in 1927 (see [30])Its further extension by the consistency-driven analysis (seethe next section aswell as [28]) can be employed as a powerfultheoretical framework for the evaluation of tenders

One of the notable pairwise comparisons past appli-cations of national importance is the evaluation of siteproposals for nuclear power plants in Holland (rejected byDutch government for details see [22]) The Monte Carlostudy presented in [26] discounts a claim of superiority ofany particular method of solving the pairwise comparisonsmatrix In fact the accuracy of solutions strongly depend onthe consistency of judgments making the consistency-drivenapproach suitable for applications

A new definition of inconsistency introduced in [28] isinstrumental for the inconsistency analysis This definitionallows us to locate the most inconsistent judgments and isinstrumental for extending the hierarchicalmodel of pairwisecomparisons by the inconsistency analysis

Issues related to public bids are documented in [32] Someof the legal aspects of bids are outlined in [33]Multiattributes(multicriteria) evaluation and its relation to the analytichierarchy process is presented by some authors None of

them however presents a comprehensive approach focusedonly on the bidding process and the evaluation of overall ofcriteria Our approach includes all the aspects

5 Example of Tendering Model

When the company already has a specific set of suppliers withwhom it intends to cooperate it should choose the best of themany suppliers characterized by various featuresThe vendorselection processmust therefore be properly designed Toolsthat help with the decision on the selection of a partnerand thus allow us to assess bids from multiple suppliers andcompare them and point to the best ones are necessary

Supplier evaluation is based on a set of both tangible andintangible criteria The former are relatively easy to compare(eg the price of a product is lower than that of anotherproduct with identical characteristics) The latter is moredifficult to survey and require detailed considerations

A practical model of a tendering process needs to beas flexible as possible (see also conclusions) Presenting anymodel for such process is risky since some readers mayconclude at this point that it is not suitable for himherOne may always discount any model as irrelevant Howeverleaving a reader without any practical application of the pre-sented framework is unacceptableTherefore a compromisedsolution is proposed and Box 1 contains a set of selectedcriteria most frequently used in tenders These criteria arebased on the indicators in the SCOR documentation [34 35]articles studies reports about selections criteria analyses ofcompanies and interviews with experts Thus our modelis a mixture of a lot of approaches to supplier selectionWe chose some criteria of supplier selection which wereverified by experts in supply chains The proposed list isnot exhaustive and does not pretend to be complete butstill fairly representative It is worthwhile to note that theauthority issuing a tender can select arbitrary criteria foreach type of bid They can also scale (or weight) particularcriteria depending on the kind (or extent) of work therequired potential of the contractor or the necessary level oftechnology

The criteria have been divided into five main groups

(i) quality of product or service (qua)(ii) flexibility and adaptability (flex)(iii) organizational potential (pot)(iv) financial standing and payment conditions (fin)(v) experience (exp)

The above-mentioned criteria together form a SupplierQuality Index (SQI)The two criteria price and delivery timeare not mentioned here deliberately These are the tangiblefactors which can be easily compared with each other butthey are usually threshold factors We could for example geta very good deal from a supplier who could deliver what weneed now in ten years For this reason our model includesfactors which are usually labeled other factors For this reasonour model should be regarded as an extension to the classicalmodel of supplier selection since the computed factor may


Tender evaluation criteriaQuality of product or service (qua)Reliability and stability of the product or service (reliability)Quality standards (q-standards)Quality guarantees (q-guaranty)Quality control system (q-system)

Flexibility and adaptability (flex)Ease of change introduction (eg of resources processes) (change)Readiness to execute orders according to clients terms (ready)Ability to deal with a changing economic environment and crisis situations(a-deal)Ability to produce personalized products or improve existing onesaccording to customersrsquo requirements (a-personal)Ability to cooperate and coordinate in a completely new product(a-cooperate)Reaction time to unexpected demand changes (reaction)

Organizational potential (pot)Production capacity (product)Logistics opportunities (logist)Experience in cooperation (cooper)Management and organizational skills (skills)

Financial standing and payment conditions (fin)Loan possibilities (loan)Cash conversion cycle (cash)Financial stability (stable)Capital rotation (capital)

Experience (exp)Number of customers (customers)Number of completed transactions (trans)Recommendation other customers (recommend)Supported industries (industry)

Box 1 Criteria taken into consideration in the selection and evaluation process of suppliers

be used on top of what we are forced to do it estimating otherfactors It is all well reflected by the satisfacing rule introducedby Herbert A Simon in 1956 (see [36]) Moreover Simon isa winner of Nobel Prize in Economics (1978) Besides thetangible factors are usually the criteria of the greatest weightin the evaluation process SQI is therefore an importantaddition to the analysis of suppliers

Obviously when choosing suppliers all types of criteriacan be considered Often they form close relationships Forexample price is inextricably linked to the quality of theproduct or service However immeasurable factors such aspayment terms and supplier flexibility have an indirect effecton the price of the product or service

The criteria presented in Box 1 may not always meet theneeds of the company Therefore they should be regardedas a starting point to determine its own set of criteria thata specific company will use This means that this list is notexhaustive and if required can be extended or shortened Itshall be noted however that the adopted criteria should beknown to all the suppliers from the enterprise network beingevaluated

For the needs of the article surveys were conductedamong managers dealing with cooperation with suppliers intheir enterprisesThe study was divided into two stages Firstthe entrepreneurs were interviewed in order to make a list ofthe most important criteria for selecting suppliers in a supplychain On this basis a questionnaire was prepared for thepreviously mentioned survey The second stage of the study(with the use of the questionnaire) was aimed at establishingthe importance of the individual criteria (by comparing themin pairs) for selecting suppliers in a supply chain

The survey was conducted in September and Octo-ber of 2013 A representative sample of 49 entrepreneursfrom Poland participated The respondents included mainlydirectors logistics specialists supply chain managers andmerchants It is worth to note that according to this formula119878119886119898119901119897119890 119904119894119911119890 = 025 times (119888119890119903119905119886119894119899119905119910 119891119886119888119905119900119903119886119888119888119890119901119905119886119887119897119890 119890119903119903119900119903)

2

in [37] showing a table with 119888119890119903119905119886119894119899119905119910 119891119886119888119905119900119903 a sample sizeof 49 observations is sufficient to achieve approximately 85certainty usually desired for a pilot study

Every person had to answer 10 questions and thusdetermine 10 ratios 119886

12 11988613 11988614 11988615 11988623 11988624 11988625 11988634 11988635 11988645


Table 2 Relative judgements for the group of criteria

qua flex pot fin expqua 1 15 12 14 13flex 07 1 08 11 09pot 09 12 1 14 12fin 07 09 07 1 11exp 08 11 09 09 1

which allows the formation of the partial PC matrix A in theform

A =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 11988612119886131198861411988615

1

11988612

1 119886231198862411988625

1

11988613

1

11988623

1 1198863411988635

1

11988614

1

11988624

1

11988634

1 11988645

1

11988615

1

11988625

1

11988635

1

11988645

1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(2)

Table 2 demonstrates a matrix with relative comparisonsA scale of 1 to 3 (and its inverse 13 to 1) is used In therepresented case the highest importance has been assignedto qua because the quality of product and service (egreliability and stability of the product or service) is usuallyof great importance to customers When compared to theflexibility and adaptability (flex) qua has been assessed tobe 15 (a compromised evaluation between 1 and 2 fromTable 1) Flexibility and adaptability are a factor relatedto the ability to deal with a changing environment andto the reaction time to unexpected demand changes It isassumed that organizational potential (pot) is less importantfor the customers (comparison of qua to pot is 12) Thisallows one to assess such evaluation criteria as productioncapacity logistics opportunities experience in cooperationmanagement and organizational skills The assessment ofthe importance of the quality factors against the financialstanding and payment conditions (fin) (eg cash conversioncycle financial stability) is set to 14 (when compared toqua) The qua factor is only 13 times more importantthan the experience (exp) which is related to the numberof customers recommendation of other customers and soforth Flex against pot is assessed to 08 which means thatpot is 12 times more important than flex (see Figure 1)The relationships between flex and fin as well as flex andexp are quite similar and equal 11 and 09 respectivelyPot against fin is set to 14 and 12 when compared to expThe last comparison is fin to exp and it equals only 11which means fin is almost equally important as exp It isworth to note that the maximum of the relative judgmentsin our research equals only 15 This means that there areno significant differences between the presented criteria Norelative judgment is absolute or even strongly important incomparison to the others

qua-flexqua-potqua-fin

qua-expflex-potflex-fin

flex-exppot-fin

pot-expfin-exp

02 04 06 08 1 12 14 160

Figure 1 Relative judgments for the group of criteria on a bar chart

Bold face 1rsquos on the main diagonal are arbitrary valuesdue to the fact that they represent a relative ratio of acriterium against itself Values below the main diagonal donot need to be entered by the user They are reciprocal tothe corresponding values in the upper triangle (for details seematrix A)

Figure 1 presents data from Table 2 on a bar chart

6 Consistency Analysis of Tendering Modeland Final Weights for Evaluated Criteria

One of the challenges posed to the pairwise comparisonsmethod is the lack of consistency of the pairwise comparisonsmatrices which arise in practice (see [38]) Given an 119899 times 119899PC matrix A which is not consistent the theory attemptsto provide a consistent 119899 times 119899 matrix B which differs frommatrix A ldquoas little as possiblerdquo Let 119904 = (119904

1 119904

119899) be the

eigenvector of A corresponding to 120590 the largest eigenvalueof A A statistical experiment (see [26]) shows howeverthat the accuracy of weights does not strongly depend onthe method In particular the geometric means method (see[39]) produced similar results with high accuracy to theten million cases There is however a strong relationshipbetween accuracy and consistency This is the main focus ofthe consistency-driven approach

In practice inconsistent judgments are difficult to avoidwhen at least three factors are independently comparedagainst each other [27] In this study consistencywas success-fully achieved For example let us look closely at the ratiosof the first three criteria in Table 2 qua (for short 119860) flex(denoted by 119861) and pot (referred to as 119862) The assessmentof 119860 against 119861 is 15 119861 against 119862 is assessed as 08 The rationof 119860 to 119862 is 12

Let us try to illustrate the inconsistency analysis From119860119861 = 15 and 119861119862 = 09 we can infer that 119860119862 = 15 sdot09 = 135 which is different from our original assessments(see Table 2 where 119860119862 = 13) In fact we do not knowwhich assessmentwas incorrect In particular (a frequent casein practice) each original assessment might have been (andusually is) only slightly inaccurate

The consistency factor (cf) is theminimumof |1minus13(15sdot09)| and |1 minus (15 sdot 09)13| which is 004 Since cf is verylow those judgments must not be reconsidered before any


Table 3 Consistency analysis

119909 119910 119911 cf15 13 09 0112 13 12 0114 13 11 0208 09 12 0011 09 11 0215 12 08 0008 11 14 0015 14 11 0108 09 12 0014 12 11 02

further calculations (eg of the final weights) can take placeFor details related to consistency analysis see [28]

Table 3 shows the consistency analysis for all triads(119860 119861 119862) from the matrix with relative comparisons (seeTable 2) where 119909 = 119860119861 119910 = 119860119862 and 119911 = 119861119862The maximum value of cf is only 02 which means thatthe respondentsrsquo answers are consistent enough based onthe former application as well as theory provided in [28]Basically the heuristic ldquoby one offrdquo and the ldquosatisficing rulerdquo(mentioned in [36]) are used Simply the comparisons do notneed to be 100 exact but they should be good enough andldquoby one offrdquo from the ideally consistentminimal cycle of threecomparisons (needed to compare three criteria which is theminimum size of a cycle two elements do not create a cycle)

By increasing or decreasing PC matrix entries (the mostinconsistent triads are easily located by the simple searchprocedure) we develop a very good orientation quickly

All computations including the final weights are doneby the Concluder software It is freely distributed by Souce-Forge as Concluder with a tutorial posted on YouTube asjConcluder It is a flexible and powerful evaluation toolfor complex systems with the expectations of having moreapplications in the future as more decisions are made undergrowing financial stress

It is not important to address all mathematical aspects ofgetting the final weights but the eigenvector method (see [28]for details) can be used to obtain results illustrated in Figure 2As we can see the product quality and service factor has thehighest weight (25) The second most important criterionis the financial standing and payment conditions (22) Theother three factors are quite similar and they equal 17-18

One common concern needs to be addressed Can we doit in short Yes you can Using a more precise consistency-driven approach may look complicated at first glance Itmay be particularly visible when it comes to making thecomparative judgments How can we start Is it not just easierto assign points to the list It is even advisable to start withassigning some points using the so-called by eye commonsense method Having that done we can easily construct thepairwise comparisons matrix A (see Section 6) by simpledivision of points for the corresponding criteria In factone may even give up at this point however after a carefulexamination of the matrix we often may be unpleasantly

surprised by our own judgments We may for examplediscover that certain ratios are surprisingly small whileothers are out of common sense limits since comparing twoat a time is easier than by eye estimation

7 Example of Suppliers Assessment

The next stage of the supplier selection is their assessmentHaving established the criteria knowing their weights onecan proceed to the analysis of suppliers in the given circum-stances If these criteria and suppliers are abundant it may behelpful to apply a multicriterion analysis using mathematicalmethods such as optimization at the so-called multiplicity ofattributes It involves calculation of the synthetic evaluationwith the use of the weighted mean in the way presentedbelow With a list of individual ratings criteria 119909

1 1199092 119909

119899

with weights 1199081 1199082 119908

119899 where 119908

119899gt 0 the weighted

mean can be calculated 119909 = sum119899119894=1119908119894119909119894sum119899

119894=1119908119894

The weighted mean is calculated for individual suppliersFinally those suppliers for which the mean value is highestare taken into account

For the needs of the paper the authors took into accounta large enterprise which produces clothes and is locatedin Poland The aim of the company was to implement asupplier evaluation process that would allow us to choosethe best fabric for the production of clothes from among thesuppliers (119860 119861 119862119863) Table 4 presents the criteria from Box 1supplemented with weights (fromFigure 2) assessments andsynthetic assessments calculated (assessment multiplied byweight) The grading scale ranges from 1 to 5 (1 being theworst rating and 5 the best) In this case the best final gradewas awarded to supplier 119863 (score 431) which despite lowerquality (lower 119909

119894value) than that of supplier 119862 received

better evaluations for most of the other criteria (higher 119909119894

value) from the recipient If supplier119863 improved its financialstanding and payment conditions and in particular the loanpossibilities cash conversion cycle and capital rotation theresult would be even better and it would reinforce its positionamong other suppliers of the analyzed company Supplier 119860got the worst evaluation (score 335) This supplier has theworst product or service quality and organizational potentialof all

The authors are aware that the scheme presented inthis section may not seem very complicated Howeverthe procedures offered by the multicriterion programmingshould be conceptually simple enough for a person whodoes not have preparation in mathematics and operationsresearch to use them without difficulty It is very importantsince the evaluation of many criteria (eg flexibility andadaptability) cannot take place without the participation ofemployees even though most of the steps in this procedureare automated

A recent application of the presented method for assess-ment at a national level in Poland is in [40]

8 Conclusions

This study presents a theoretical framework for building alogicalmodel for evaluation of suppliers It also proposes how


Table 4 Sample assignment of numerical values to criteria for supplier evaluation

Evaluation criterion Weight Sup 119860 Sup 119861 Sup 119862 Sup119863119908119894

11990911199081lowast 119909111990921199082lowast 119909211990931199083lowast 119909311990941199084lowast 1199094

Quality of product or service 025 3 075 4 1 5 125 4 1Organizational potential 018 2 036 4 072 5 09 5 09Financial standing 022 3 066 4 088 3 066 3 066Experience 017 4 068 4 068 4 068 5 085Flexibility and adaptability 018 5 09 5 09 4 072 5 09Total 100 mdash 335 mdash 418 mdash 421 mdash 431

Criterium Bar graph representationR-totWeightname

(1) Product or service quality 25(2) Financial standing 22(3) Organizational potential 18(4) Flexibility and adaptability 18(5) Experience 17

25476583

100

Where R-tot means running total of the weight

Figure 2 The final weights for the evaluated criteria

the final weightsmay be obtained from relative pairwise com-parisons However the framework is not a ready cookbookfor constructing such models Models can be constructed inmany different ways A goodmodel should however containboth tangible and intangible criteria Identification of majorcriteria is one of the essential components of constructingmodels Each model strongly depends on the specifics ofthe given case and its business environment The presentedapproach allows the buyers to create their own list of criteriawithout any limitations

The price and delivery time have not been included in ourapproachThey are threshold factors and reserved for the finaldecision by the evaluation panel In other words the modelprovides the assessments of tenders as a vector of weightsThese weights are applied to evaluation of each individualproposalThefinal product of the evaluation generates a list oftenders with the overall score (received by a vector product ofweights and corresponding evaluations for each criteria) plusthe price and the delivery time Frequently it is an evaluationpanelrsquos decision whether or not to award the contract to thebest tender according to the overall score

There are easy cases when the winnerrsquos proposed priceand delivery time are similar to the next but in other casesit may be substantially different Simply the financial con-straints may prohibit consideration of some solutions (egthe delivery time may be unacceptable) The consistency-driven pairwise comparisons method can be used by every-one on a personal computer thanks to aforementionedConcluder software

More research is needed to shortlist all intangible criteriawhich are used to evaluate suppliers Currently it is done bythe intuition but it needs to be investigated The proposedmodel can process any number of such criteria

The authors intend to carry out studies on a larger sampleof managers who will represent not only Polish companies

but also foreign onesThis will allow us to compare the resultsand highlight potential differences in the supplier evaluationcriteria In addition it will increase the representativeness ofthe research and hence the results will be more universal

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The paper was written with the financial support from theNational Center of Science Grant no DEC-201103DHS403367

References

[1] Z G Zacharia NWNix andR F Lusch ldquoAn analysis of supplychain collaborations and their effect on performance outcomesrdquoJournal of Business Logistics vol 30 no 2 pp 101ndash123 2009

[2] H Chen P J Daugherty and A S Roath ldquoDefining andoperationalizing supply chain process integrationrdquo Journal ofBusiness Logistics vol 30 no 1 pp 63ndash84 2009

[3] M Anholcer and A Kawa ldquoOptimization of supply chain viareduction of complaints ratiordquo in Technologies and ApplicationsG Jezic M Kusek N T Nguyen R J Howlett and L C JainEds pp 622ndash628 Springer Berlin Germany 2012

[4] A Kawa ldquoSMART logistics chainrdquo in Intelligent Informationand Database Systems pp 432ndash438 Springer Berlin Germany2012

[5] A Zouggari and L Benyoucef ldquoSimulation based fuzzy TOPSISapproach for group multi-criteria supplier selection problemrdquoEngineering Applications of Artificial Intelligence vol 25 no 3pp 507ndash519 2012


[6] E Hofmann and H Kotzab ldquoA supply chain-oriented approachof working capital managementrdquo Journal of Business Logisticsvol 31 no 2 pp 305ndash330 2010

[7] S Tully ldquoPurchasings new musclerdquo Fortune vol 131 no 3 pp75ndash84 1995

[8] T T Burton ldquoJITrepetitive sourcing strategies tying the knotwith your suppliersrdquo Production and Inventory ManagementJournal vol 29 no 4 pp 38ndash41 1988

[9] R J Vokurka J Choobineh and L Vadi ldquoA prototype expertsystem for the evaluation and selection of potential suppliersrdquoInternational Journal of Operations and Production Manage-ment vol 16 no 12 pp 106ndash127 1996

[10] B Ageron A Gunasekaran and A Spalanzani ldquoISIT assupplier selection criterion for upstream value chainrdquo IndustrialManagement and Data Systems vol 113 no 3 pp 443ndash4602013

[11] M Leenders P F Johnson A Flynn and H E FearonPurchasing Supply Management McGraw-Hill New York NYUSA 13th edition 2006

[12] R M Monczka R Trent and R Handfield Purchasing andSupply ChainManagement South-WesternThomson LearningCincinnati Ohio USA 2nd edition 2002

[13] A Herbon S Moalem H Shnaiderman and J TemplemanldquoDynamic weights approach for off-line sequencing of supplierselection over a finite planning horizonrdquo International Journalof Physical Distribution and Logistics Management vol 42 no5 pp 434ndash463 2012

[14] J A Ogden ldquoSupply base reduction an empirical study ofcritical success factorsrdquo Journal of Supply Chain Managementvol 42 no 4 pp 29ndash39 2006

[15] A Sarkar and P K J Mohapatra ldquoEvaluation of supplier capa-bility and performance a method for supply base reductionrdquoJournal of Purchasing and Supply Management vol 12 no 3 pp148ndash163 2006

[16] T Y Choi andDRKrause ldquoThe supply base and its complexityimplications for transaction costs risks responsiveness andinnovationrdquo Journal of Operations Management vol 24 no 5pp 637ndash652 2006

[17] P D Cousins ldquoSupply base rationalisation myth or realityrdquoEuropean Journal of Purchasing and Supply Management vol 5no 3-4 pp 143ndash155 1999

[18] I Igoulalene L Benyoucef and M K Tiwari ldquoNovel fuzzyhybrid multi-criteria group decision making approaches forthe strategic supplier selection problemrdquo Expert Systems withApplications vol 42 no 7 pp 3342ndash3346 2015

[19] X Zhang Y Deng F T Chan and S Mahadevan ldquoA fuzzyextended analytic network process-based approach for globalsupplier selectionrdquo Applied Intelligence vol 43 no 4 pp 760ndash772 2015

[20] A Ishizaka ldquoComparison of fuzzy logic AHP FAHPandhybridfuzzy AHP for new supplier selection and its performance anal-ysisrdquo International Journal of Integrated Supply Managementvol 9 no 1-2 pp 1ndash22 2014

[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] P Nijkamp P Rietvield and H VoogdMulticriteria Evaluationin Physical Planning North-Holland Amsterdam The Nether-lands 1990

[23] H Voogd Multicriteria Evaluation for Urban and RegionalPlanning Pion London UK 1983

[24] H Ching-Lai and Y Kwangsun Multiply Attribute DecisionMaking Methods and Applications Springer Berlin Germany1981

[25] J FulopWWKoczkodaj and S Szarek ldquoAdifferent perspectiveon a scale for pairwise comparisonsrdquo in Transactions on Com-putational Collective Intelligence I pp 71ndash84 Springer BerlinGermany 2010

[26] M Herman and W W Koczkodaj ldquoMonte carlo study ofpairwise comparisonsrdquo Information Processing Letters vol 57no 1 pp 25ndash29 1996

[27] W W Koczkodaj and R Szwarc ldquoOn axiomatization of incon-sistency indicators for pairwise comparisonsrdquo FundamentaInformaticae vol 132 no 4 pp 485ndash500 2014

[28] W W Koczkodaj ldquoA new definition of consistency of pairwisecomparisonsrdquo Mathematical and Computer Modelling vol 18no 7 pp 79ndash84 1993

[29] Z Duszak and W W Koczkodaj ldquoGeneralization of a new def-inition of consistency for pairwise comparisonsrdquo InformationProcessing Letters vol 52 no 5 pp 273ndash276 1994

[30] L LThurstone ldquoA law of comparative judgmentrdquo PsychologicalReview vol 34 no 4 pp 273ndash286 1927

[31] W W Koczkodaj ldquoStatistically accurate evidence of improvederror rate by pairwise comparisonsrdquoPerceptual andMotor Skillsvol 82 no 1 pp 43ndash48 1996

[32] H Al-Tabtabai and J Diekmann ldquoJudgemental forecastingin construction projectsrdquo Construction Management and Eco-nomics vol 10 no 1 pp 19ndash30 1992

[33] L G Crowley and D E Hancher ldquoEvaluation of competitivebidsrdquo Journal of Construction Engineering andManagement vol121 no 2 pp 238ndash243 1995

[34] P Bolstroff and R G Rosenbaum Supply Chain Excellence aHandbook for Dramatic Improvement Using the SCOR Model(Vol 1) Amacom New York NY USA 2007

[35] Supply Chain Council SCOR Supply Chain Operations Refer-ence Model Ver 100 Supply Chain Council Washington DCUSA 2010

[36] H A Simon ldquoRational choice and the structure of the environ-mentrdquo Psychological Review vol 63 no 2 pp 129ndash138 1956

[37] JWhitten and L Bentley Systems Analysis and DesignMethodsMcGraw-Hill Professional 2005

[38] C Lin G Kou and D Ergu ldquoA statistical approach to measurethe consistency level of the pairwise comparison matrixrdquoJournal of the Operational Research Society vol 65 no 9 pp1380ndash1386 2014

[39] R E Jensen ldquoAn alternative scaling method for priorities inhierarchical structuresrdquo Journal of Mathematical Psychologyvol 28 no 3 pp 317ndash332 1984

[40] W W Koczkodaj K Kułakowski and A Ligeza ldquoOn thequality evaluation of scientific entities in Poland supported byconsistency-driven pairwise comparisonsmethodrdquo Scientomet-rics vol 99 no 3 pp 911ndash926 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


selection of suppliers for an organization has been stressedin the literature [11 12]

The supplier selection process requires predeterminationof several issues including the number of suppliers withwhich the organization wishes to contract for a given productor service [13] It is difficult to determine howmany suppliersshould work for a given organization Some authors arguethat there is now a trend to reduce the number of suppliers toa manageable level [14 15] even to a single source in extremecases

No clear recommendation regarding the number of sup-pliers is confirmed by the results of the research Every thirdcompanydoes not follow any general ruleThis is understand-able On the one hand a large number of suppliers ensureslower prices for customers offers greater safety and decreasesthe risk of stopping production On the other hand it raisesthe operating costs of such cooperation (maintenance costsof information systems controlling sourcing negotiationsetting conditions for the cooperation audits etc) [16 17]

In the process of partner selection several methodsare used Among them categorical method weighted-pointmethod vendor performance matrix approach vendor pro-file analysis (VPA) multicriteria decision aid (MCDA) mul-tiple objective programming (MOP) such as goal program-ming data envelopment analysis (DEA) and multiattributeutility theory (MAUT) The recent studies about the useof fuzzy theories and their development in the supplierselection problem are particularly interesting [18ndash20] In [21]the authors proposed a new tool for supplier selection basedon a combination of the grey system and the uncertaintytheory neither of which requires any probability distributionor fuzzy membership function

The structure of the paper is as follows the characteristicof bidding process is discussed in Section 2 Then Sec-tion 3 describes supplier evaluation problems and Section 4presents pairwise comparisons method and review of theother applications and the literature on the pairwise com-parisons Section 5 provides example of a tendering modelproposed by the authors Next Section 6 contains consistencyanalysis of tendering model Section 7 proposes an exampleof suppliers assessment The conclusions of research areoutlined in Section 8

2 Bidding Process

The previously mentioned methods have some limitationsbecause the supplier selection process is mostly based onintuitionThere is no theoretical base or consistentmethod ofpredicting the best bid It is not uncommon for the evaluationpanel to arrive at a deadlock when a part of the panel favorsone solution because of certain criteria while the otherpart insists on another solution since it scores better ondifferent criteria The decision making process nearly alwaysinvolves some kind of constituency in modern democraticsocieties We have various boards of governors or directorscommittees task groups city councils panels of experts andindividuals each with a specific agenda Heated discussionsand various ways of dispute and argumentation often takeplace to arrive at certain decisions

Most constituencies have worked out precise and practi-cal policies for running meetings in an orderly and effectivemanner What we lack however is a device for drawing solidconsistent conclusions and all too often the loudest individualwins Unfortunately loudness does not necessarily go alongwith wisdom Casual thinking is not efficient in predictingcomplex outcomes

The main goal of tendering is the selection of the mostsuitable supplier from the company point of view Through abidding we try to achieve the following

(i) the setting of common input constraints for potentialsuppliers

(ii) the selection of the best supplier based on tangibleand intangible but constant (during the entire biddingprocess) criteria which allow us to compare theproposed offers

(iii) a minimization of the influence of informal interestson selection of an offer thanks to the application of astrict selection process

One of the most instrumental conditions of a fair biddingis the necessity of a precise scoring system of all criteriaand their preferences in the tendering documents to be usedduring the selection process by the selection panel

A selection panel is obligated to use objective and mea-surable evaluation criteria during the selection process Thegoal of such an action is to have more comparability andfewer intangible or arbitrary decisions Application of suchquantification in the request for proposals (RFPs) to specificbidding components together with the bidding amounthelps in the selection of the best offer A recommendedpractice is the assignment of weights (eg factors which arepositive points) to each aspect of an offer with respect to thecriteria listed in the bidding documents As a result the bestoffer is the one with the highest total score

In restricted bidding (as opposed to open bidding) onlyinvited parties may submit a bid In the first stage of atwo-stage bidding process suppliers submit preliminary bidswithout the total amount However the selection panel mayrequest precise specification of some parameters Successfulbidders from the first stage are invited to submit a completebid for the second stage Two-stage bidding may be consid-ered a special case of an open bidding The main differencesare as follows

(i) the possibility of submitting two tendering offers bybidders

(ii) the use of two selections instead of one the firstselection to prequalify tenders and the second to findthe true winner

(iii) the possibility (and sometimes necessity) of negotia-tions with bidders

(iv) the possibility of using the negotiation results tochange or restrain crucial constraints of the requestfor proposals before they are distributed again tobidders selected during the first stage of bidding


















the literature

A =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 11988612sdot sdot sdot 1198861119899

1

11988612


1

1198861119899

1

1198862119899

sdot sdot sdot 1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(1)


119904119894 over 119904

119895

Coefficients 119886119894119895


119894119894= 1 119886



119894119895) such that 119886

119894119895gt 0 for every





119894119896= 119886119894119895sdot 119886119895119896


1 119904




119894119895) there


119894119895= 119904119894119904119895for







1 Equal importance





14 and soforth




























2



12 11988613 11988614 11988615 11988623 11988624 11988625 11988634 11988635 11988645





A =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 11988612119886131198861411988615

1

11988612

1 119886231198862411988625

1

11988613

1

11988623

1 1198863411988635

1

11988614

1

11988624

1

11988634

1 11988645

1

11988615

1

11988625

1

11988635

1

11988645

1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(2)




flex-exppot-fin

pot-expfin-exp

02 04 06 08 1 12 14 160






1 119904

119899) be the







119909 119910 119911 cf15 13 09 0112 13 12 0114 13 11 0208 09 12 0011 09 11 0215 12 08 0008 11 14 0015 14 11 0108 09 12 0014 12 11 02










1 1199092 119909

119899

with weights 1199081 1199082 119908

119899 where 119908



119894=1119908119894








8 Conclusions









25476583

100











Acknowledgment


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























the literature

A =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 11988612sdot sdot sdot 1198861119899

1

11988612


1

1198861119899

1

1198862119899

sdot sdot sdot 1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(1)


119904119894 over 119904

119895

Coefficients 119886119894119895


119894119894= 1 119886



119894119895) such that 119886

119894119895gt 0 for every





119894119896= 119886119894119895sdot 119886119895119896


1 119904




119894119895) there


119894119895= 119904119894119904119895for







1 Equal importance





14 and soforth




























2



12 11988613 11988614 11988615 11988623 11988624 11988625 11988634 11988635 11988645





A =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 11988612119886131198861411988615

1

11988612

1 119886231198862411988625

1

11988613

1

11988623

1 1198863411988635

1

11988614

1

11988624

1

11988634

1 11988645

1

11988615

1

11988625

1

11988635

1

11988645

1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(2)




flex-exppot-fin

pot-expfin-exp

02 04 06 08 1 12 14 160






1 119904

119899) be the







119909 119910 119911 cf15 13 09 0112 13 12 0114 13 11 0208 09 12 0011 09 11 0215 12 08 0008 11 14 0015 14 11 0108 09 12 0014 12 11 02










1 1199092 119909

119899

with weights 1199081 1199082 119908

119899 where 119908



119894=1119908119894








8 Conclusions









25476583

100











Acknowledgment


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1 Equal importance





14 and soforth




























2



12 11988613 11988614 11988615 11988623 11988624 11988625 11988634 11988635 11988645





A =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 11988612119886131198861411988615

1

11988612

1 119886231198862411988625

1

11988613

1

11988623

1 1198863411988635

1

11988614

1

11988624

1

11988634

1 11988645

1

11988615

1

11988625

1

11988635

1

11988645

1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(2)




flex-exppot-fin

pot-expfin-exp

02 04 06 08 1 12 14 160






1 119904

119899) be the







119909 119910 119911 cf15 13 09 0112 13 12 0114 13 11 0208 09 12 0011 09 11 0215 12 08 0008 11 14 0015 14 11 0108 09 12 0014 12 11 02










1 1199092 119909

119899

with weights 1199081 1199082 119908

119899 where 119908



119894=1119908119894








8 Conclusions









25476583

100











Acknowledgment


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















2



12 11988613 11988614 11988615 11988623 11988624 11988625 11988634 11988635 11988645





A =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 11988612119886131198861411988615

1

11988612

1 119886231198862411988625

1

11988613

1

11988623

1 1198863411988635

1

11988614

1

11988624

1

11988634

1 11988645

1

11988615

1

11988625

1

11988635

1

11988645

1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(2)




flex-exppot-fin

pot-expfin-exp

02 04 06 08 1 12 14 160






1 119904

119899) be the







119909 119910 119911 cf15 13 09 0112 13 12 0114 13 11 0208 09 12 0011 09 11 0215 12 08 0008 11 14 0015 14 11 0108 09 12 0014 12 11 02










1 1199092 119909

119899

with weights 1199081 1199082 119908

119899 where 119908



119894=1119908119894








8 Conclusions









25476583

100











Acknowledgment


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













A =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 11988612119886131198861411988615

1

11988612

1 119886231198862411988625

1

11988613

1

11988623

1 1198863411988635

1

11988614

1

11988624

1

11988634

1 11988645

1

11988615

1

11988625

1

11988635

1

11988645

1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(2)




flex-exppot-fin

pot-expfin-exp

02 04 06 08 1 12 14 160






1 119904

119899) be the







119909 119910 119911 cf15 13 09 0112 13 12 0114 13 11 0208 09 12 0011 09 11 0215 12 08 0008 11 14 0015 14 11 0108 09 12 0014 12 11 02










1 1199092 119909

119899

with weights 1199081 1199082 119908

119899 where 119908



119894=1119908119894








8 Conclusions









25476583

100











Acknowledgment


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909 119910 119911 cf15 13 09 0112 13 12 0114 13 11 0208 09 12 0011 09 11 0215 12 08 0008 11 14 0015 14 11 0108 09 12 0014 12 11 02










1 1199092 119909

119899

with weights 1199081 1199082 119908

119899 where 119908



119894=1119908119894








8 Conclusions









25476583

100











Acknowledgment


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















25476583

100











Acknowledgment


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Supplier Evaluation Process by Pairwise...

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