A Novel Contractor Selection Technique Using the Extended ...

Research ArticleA Novel Contractor Selection Technique Using the ExtendedPROMETHEE II Method

Kuei-Hu Chang 12

1Department of Management Sciences Chinese Military Academy Kaohsiung 830 Taiwan2Institute of Innovation and Circular Economy Asia University Taichung 413 Taiwan

Correspondence should be addressed to Kuei-Hu Chang evenken2002gmailcom

Received 25 September 2021 Revised 26 October 2021 Accepted 1 November 2021 Published 18 November 2021

Academic Editor Ali Ahmadian

Copyright copy 2021 Kuei-Hu Chang )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

Selecting suitable contractors directly influences product quality corporate profits and even sustainable development )eselection problem of contractors is therefore a critical issue for the sustainable development of an enterprise However tra-ditional contractor selection techniques are unable to handle information regarding the relative importance of criteria or handlenonexistent or missing data in the assessment process of contractor selection In order to effectively address this problem thisstudy proposes a new contractor selection technique that integrates the concept of soft set and the PROMETHEE II method toselect suitable contractors )ree numerical examples are applied to prove the correctness and effectiveness of the proposedtechnique )is study also compares the simulation results achieved using the proposed method with those achieved using thetraditional weighted arithmetic averaging method and the data envelopment analysis (DEA) technique )e simulation resultsshow that the proposed method is a more general contractor selection technique for handling incomplete information than thetraditional weighted arithmetic averaging method and the DEA method

1 Introduction

Contractor selection includes multiple performance as-sessment criteria and is a multicriteria decision-making(MCDM) issue Choosing suitable contractors directly af-fects the competitive advantage of products and the sus-tainable development of enterprises )erefore contractorselection is a critical issue in the supply chain and has re-ceived considerable research attention Many authors haveused different computation methods to address contractorselection problems For example San Cristobal [1] combinedVlseKriterijumska Optimizacija I Kompromisno Resenje(VIKOR) techniques for order preference by similarity toideal solution (TOPSIS) and the analytical hierarchy process(AHP) methods for selecting suitable contractors for the ldquoLaBraguıardquo road-building project in Spain Yang et al [2]proposed an approach based on data envelopment analysis(DEA) that was applied to support the selection of best valuecontractors in single-input and multiple-output mannersAkcay et al [3] used the concept of fuzzy logic to propose a

fuzzy decision support model for contractor selection ofTurkish construction projects Hasnain et al [4] proposed ananalytical network process (ANP) based decision supportsystem to solve the most valuable contractor selectionproblems in road construction projects Chang [5] inte-grated the soft set and intuitionistic fuzzy weighted averagemethods to select the best supplier under an incompleteinformation environment Gharedaghi and Omidvari [6]integrated the analytical network processing decision-making trial and evaluation laboratory (DEMATEL) andthe grey system theory to select suitable safety contractorsfor oil and gas industries

)e contractor selection assessment process may en-counter instances of missing or nonexistent assessmentcriteria data Incomplete attribute value information in-creases the difficulty of assessing contractors In order tohandle incomplete information the traditional contractorselection method directly deletes incomplete attribute valueinformation Deleting information however results in areduction in available information and distorts evaluation

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 3664709 11 pageshttpsdoiorg10115520213664709

results Fortunately the soft set method is able to handleincomplete assessment criteria information )e soft setmethod was first proposed by Molodtsov [7] to handle theuncertainty of data or traditional mathematical tools unableto handle fuzzy information in the process of informationfusion and following the development of soft sets numerousstudies used the soft set method to handle different decision-making problems [7ndash20]

Contractors play a key role in successful operations andoverall project performance in addition choosing the rightcontractors for a particular project is a major challenge in thesupply chain However traditional contractor selectionmethods do not consider information regarding the relativeimportance of criteria )e preference ranking organizationmethod for enrichment evaluation (PROMETHEE) methodis one of the most commonly used techniques to solveMCDM issues)e advantages of the PROMETHEEmethodare that the relative importance of criteria is consideredPROMETHEE method was first introduced by Brans andVincke [21] applied as an outranking relation techniquebetween pairs of alternatives to solve MCDM problemsBecause the PROMETHEE methodrsquos calculation is simpleand it is easy to operate numerous studies have used thismethod to handle decision-making problems in variousfields For example Brankovic et al [22] used PROMETHEEand the simplified elimination and choice translating reality(ELECTRE) approach to determine the criterion weight forselecting optimal alternative hydraulic structure solutionsTian et al [23] proposed an improved PROMETHEE IImethod based on axiomatic fuzzy set theory which si-multaneously considered the subjective preferences of ex-perts and the objective weights of assessment criteria )eimproved PROMETHEE II method considers ranking aswell as the degree of credibility of the raw data )ePROMETHEEmethod has been used in many fields such asalternative locations for solar power plants [24] the as-sembly line sequencing problem [25] renewable energysource assessment [26 27] credit risk assessment [28]electricity distribution utility performance assessment [29]logistics warehouse location selection [30] petrochemicalindustry [31] occupational health and safety [32] andcybersecurity of Industry 40 [33] However the typicalPROMETHEE method cannot handle the incomplete as-sessment criteria information

Recently Yang et al [2] introduced a DEA-based ap-proach to support the selection of the best value contractorHowever this approach would cause a high repetition rateproblem when DEA values are 100 Moreover this ap-proach cannot handle cases when the needed expert data ismissing or nonexistent during the information assessmentprocess In order to effectively resolve the above contractorselection issues this study proposes a flexible PROMETHEEII method to deal with the contractor selection issue whichsimultaneously includes complete and incomplete infor-mation To verify the effectiveness of the proposed flexiblePROMETHEE II approach three contractor selection caseswere adopted and the simulation results were comparedwith the traditional weighted arithmetic averaging methodand the DEA method [2]

)e major contributions of this paper include the fol-lowing advantages (1) the proposed approach can handleincomplete assessment criteria information (2) the pro-posed approach considers the relative importance of criteria(3) the proposed approach considers the subjective pref-erences of experts (4) the traditional weighted arithmeticaverage approach and DEA method can be viewed as specialcases of the proposed approach and (5) the proposed ap-proach provides a more flexible contractor selection tech-nique to support contractor assessment for selection

)e remainder of this paper is arranged as followsSection 2 reviews related research on the PROMETHEE IImethod and soft set method Section 3 introduces theproposed novel contractor selection approach for solvingbest-value tendering selection problems )ree contractorselection case projects are adopted and the comparison withother related methods are discussed in Section 4 Finally theconclusion is given in Section 5

2 Preliminaries

)is section presents some fundamental definitions andconcepts related to the PROMETHEE II method and soft set

21 PROMETHEE II Method Brans and Vincke [21] pro-posed the PROMETHEE II approach that based on thepairwise comparison of alternatives for each criterion tosolve MCDM problems Two types of information are re-quired for the PROMETHEE II approach (1) the relativeimportance of the criteria and (2) decision-makerrsquos pref-erence function for comparing alternative contributions[34 35]

)e PROMETHEE II method consists of five steps [36]

Step 1 normalize the decision matrix

Rij xij minus min xij1113872 11138731113960 1113961

max xij1113872 1113873 minus min xij1113872 1113873(i 1 2 n j 1 2 m)

(1)

where xij is the performance index of the ith alternativeof the jth criterionFor nonbeneficial criteria equation (1) can be rewrittenas follows

Rij max xij1113872 1113873 minus xij1113960 1113961

max xij1113872 1113873 minus min xij1113872 1113873i 1 2 n j 1 2 m

(2)

Step 2 calculate the preference function Pj(a b) asfollows

Pj(a b) 0 if Raj leRbj (3)

Pj(a b) Raj minus Rbj1113872 1113873 if Raj gtRbj (4)

Step 3 calculate the aggregated preference functionπ(a b) as follows

2 Mathematical Problems in Engineering

π(a b) 1113944m

j1wjPj(a b) (5)

where wj is the weight of the jth criterionStep 4 calculate the entering flow leaving flow and netflow for each alternative

φminus(a)

1m minus 1

1113944

m

b1π(b a) (6)

φ+(a)

1m minus 1

1113944

m

b1π(a b) (7)

φ(a) φ+(a) minus φminus

(a) (8)

where φminus (a) φ+(a) and φ(a) represent the enteringflow leaving flow and net flow for each alternativerespectivelyStep 5 determine the ranking of all the possiblealternatives

All possible alternatives are ranked according to the netflow value φ(a) with a higher net flow value φ(a) repre-senting a better alternative

22 Soft Set )e soft set method is a novel mathematicalmethod developed by Molodtsov [7] to handle the relatedissues of uncertain and ambiguous data and the method isexplained as follows assume U indicates an initial universeand E be the parameters set related to the objects in U )eset P(U) be the power set of U and AsubeE

Definition 1 [7 37] A pair (F A) is called a soft set (on U)where F is a mapping given by F A⟶ P(U)

In other words the soft set over U is a parameterizedfamily of universe U subsets

Definition 2 [7 38] Both two soft sets (F X) and (G Y) arein a common universe U and (F X) is the soft subset of (GY) represented as (F X) 1113958sub (G Y) if

(i) XsubeY and(ii) foralle isinX F(e)subeG(e)

Definition 3 [38 39] Both two soft sets (F X) and (G Y) arein a common universe U where the union of (F X) and (GY) is represented as (H Z) and the following conditions aresatisfied

(i) Z XcupY and(ii) foralle isinZ H(e)

F(e) if e isin X minus Y

G(e) if e isin Y minus X

F(e)cupG(e) if e isin XcapY

⎧⎪⎨

⎪⎩

Definition 4 [38 39] Both two soft sets (F X) and (G Y) arein a common universe U where the intersection of (F X)and (G Y) is represented as (H Z) and the followingconditions are satisfied

(i) Z XcapY and(ii) foralle isinZ H(e) F(e) or G(e)

3 The Proposed Novel ContractorSelection Method

Government procurement in Taiwan involves two prin-ciples for contractor selection the best value bid and thelowest bid )e assessment criteria for the best value bidsimultaneously include qualitative and quantitative dataand present a complicated MCDM problem Howeverthe traditional contractor selection approach does notconsider the relative importance of criteria or subjectivepreferences of experts Moreover instances may occur inthe contractor selection assessment process in whichassessment criteria data are missing or nonexistent )iswill make the contractor selection evaluation morecomplicated and difficult In order to effectively addressthis problem this paper proposes a flexible PROM-ETHEE II method to deal with contractor selectionproblems which simultaneously includes complete andincomplete information )is study uses the weightedarithmetic averaging method to fill in incomplete as-sessment criteria score data Furthermore this studyapplies the PROMETHEE II method to handle infor-mation about the relative importance of criteria andsubjective preferences of experts )is study integratesthe PROMETHEE II method and soft set method to selectsuitable contractors )e results are more suitably andflexibly reflect the actual situation )e implementationsteps of the proposed novel contractor selection approachare as follows (depicted in Figure 1)

Step 1 confirm the bidding document assessmentcriteriaDetermine the content of assessment criteria accordingto the requirements of the actual bidding documentsStep 2 determine the relative weights of the assessmentcriteriaetermine the relative weights of the assessment criteriaaccording to the importance level of the assessmentcriteriaStep 3 determine the scores of the assessment criteriafor each bidderEach committee member determines the assessmentcriteria scores for each bidderStep 4 fill in incomplete assessment criteria score dataIf the assessment criteria scores are incomplete use theweighted arithmetic average approach to fill in in-complete assessment criteria score dataStep 5 normalize the decision matrixUse equations (1) and (2) to compute the normalizeddecision matrixStep 6 compute the preference function Pj(a b)Use equations (3) and (4) to compute the preferencefunction Pj(a b)

Mathematical Problems in Engineering 3

Step 7 compute the aggregated preference functionπ(a b)Use equation (5) to compute the aggregated preferencefunction π(a b)Step 8 compute the entering flow and leaving flow foreach alternativeUse equations (6)ndash(7) to compute the entering flow andleaving flow for each alternativeStep 9 identify the ranking of all possible alternativesUse equation (8) to compute the net flow values foreach alternative Rank the net flow φ(a) from highest tolowest with a higher net flow value φ(a) representing abetter bidder

4 Illustrative Example

41 Case Project 1 )is section presents an illustrative ex-ample of software supplier selection adapted from Yang et al[2] who proved the effectiveness of the proposed novelcontractor selection approach)e case companymust selectthe optimal software supplier in order to develop and planthe management information system )ere were six bid-ders and a bid opening committee that consisted of fourcommittee members in this software supplier case )eassessment criteria included the comprehension degree forthe project (C1) system design and planning (C2) codingability of a system (C3) management ability of project (C4)and past achievements and experience (C5) )e relativeweights of the five evaluation criteria were 010 030 030020 and 010 respectively )e five evaluation criteria were

then used by the final bid opening committee to determinethe most suitable software supplier )e five assessmentcriteria scores for each bidder are listed in Table 1

411 Solution by the Traditional Weighted Arithmetic Av-eraging Approach )e traditional weighted arithmetic av-eraging approach is one of the simplest and most commonlyused aggregation operators and it can be explained asfollows

Definition 5 [40 41] A weighted arithmetic averaging(WAA) operator of dimension n is a mapping F Rn⟶ R

with associated weight vector W (w1 w2 wn) and1113936

ni1 wi 1 wi isin [0 1] as follows

WWA a1 a2 an( 1113857 1113944 ni1wiai (9)

where ai is the argument variableAccording to the results of Table 1 equation (9) was used

to calculate the bidder scores for software supplier selectionas shown in Table 2

412 Solution by the DEA Method Yang et al [2] proposeda DEAmodel with single input and multiple outputs to solvethe best-value contractor selection problem )ey used thedecision-making unit (DMU) as a bidder and the DEAinput values were equal to 1 According to Table 1 the DEAPsoftware was used to run the DEA CCR model and thesoftware supplier evaluation results as listed in Table 3where a bidderrsquos DEA value of 100 indicates that the bidderhas the highest comparative efficiency

413 Solution by the Proposed Novel Contractor SelectionApproach )e proposed novel contractor selection ap-proach integrates the PROMETHEE II method and the softset approach to select suitable contractors In the softwaresupplier selection example the five assessment criteriascores for six bidders in the final bid opening committee areas listed in Table 1 (Steps 1sim 4) )e following proceduredescribes the remaining steps of the proposed method

Step 5 normalize the decision matrixAccording to Table 1 equations (1) and (2) were used tocalculate the normalized decision matrix for differentsoftware suppliers as listed in Table 4Step 6 compute the preference function Pj(a b)Step 7 compute the aggregated preference functionπ(a b)According to Table 4 equations (3)ndash(5) were used tocompute the preference function (Pj(a b)) and ag-gregated preference function (π(a b)) for differentsoftware suppliers as listed in Table 5Step 8 compute the entering flow and leaving flow foreach alternativeAccording to Table 5 equations (6) and (7) were used tocalculate the entering and leaving flows for differentsoftware suppliers as shown in Table 6

Confirm the bidding document assessment criteria

Determine the relative weights of the assessment criteria

Determine the scores of the assessment criteria for each bidder

Fill in incomplete assessment criteria score data

Normalize the decision matrix

Compute the preference function

Compute the aggregated preference function

Compute the entering flow leaving flow for each alternative

Identify the ranking of all possible alternatives

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Step 7

Step 8

Step 9

Figure 1 Flowchart of the proposed novel contractor selectionmethod


Step 9 determine the ranking of all possiblealternatives

According to Table 6 equation (8) was used to calculatethe net flow for different software suppliers )e net flowvalues φ(a) was ranked from highest to lowest with a highernet flow value φ(a) representing a better software supplieras listed in Table 7

414 Summary Case Project 1 involves selecting a softwaresupplier to develop a management information system forthe case company )e input data for each bidder was listedin Table 1 Table 8 lists the different calculation results of thetraditional weighted arithmetic averaging method the DEAmethod and the proposed method for Case Project 1

From Table 8 the DEA values for bidders 3 and 5 were100 )e proposed extended PROMETHEE II method cansolve this problem of duplicate DEA value In the proposedmethod the net outranking flow of bidders 3 and 5 were0490 and 0511 respectively )erefore the proposedmethod can better sort bidder scores

42 Case Project 2 Case Project 2 involves the selection oftwo security companies to provide security services for ascience park [2] )ere were seven bidders competing fortwo awards in this case )e assessment criteria includedcompany organization (C1) planning feasibility (C2)professional capability (C3) price (C4) and presentationand question response (C5) )e relative weights for the fiveassessment criteria were 020 025 025 020 and 010respectively and the final evaluation committee selected thebest two security services companies based on the five as-sessment criteria )e five assessment criteria scores for eachsecurity services company are listed in Table 9

421 Solution by the Traditional Weighted Arithmetic Av-erage Approach Based on the results of Table 9 equation (9)was used to calculate the bidder scores for security servicescompany selection as shown in Table 10

422 Solution by the DEAMethod According to the data inTable 9 the DEAP software was used to run the DEA CCRmodel )e security services company selection evaluationresults are listed in Table 11 In Table 11 a bidderrsquos DEAvalue of 100 indicates that the bidderrsquos performance is thehighest

423 Solution by the Proposed Method According to theresults of Table 9 equations (1) and (2) were used to calculatethe normalized decision matrix of different security com-panies as listed in Table 12

According to the results of Table 12 equations (3)ndash(5)were used to calculate the preference function Pj(a b) andaggregated preference function for different security com-panies as shown in Table 13

According to the results of Table 13 equations (6)ndash(8)were used to calculate the entering flow leaving flow and netoutranking flow for different security companies as listed inTable 14

424 Summary Case Project 2 involved the selection of twosecurity companies to provide security for a science park inTaiwan Table 15 displays the calculation results of thetraditional weighted arithmetic averaging method the DEAmethod and the proposed method for Case Project 2

In Table 15 the traditional weighted arithmetic aver-aging method identifies bidders 1 and 6 as having the twohighest scores thus the winning bidders However thetraditional weighted arithmetic averaging method does notconsider the relative importance of the criteria or subjectivepreferences of the experts which will result in an incorrectconclusion From Table 15 the DEA values for bidders 1 23 6 and 7 are 100 )e DEA method thus has a highduplicate rate problem )e proposed extended PROM-ETHEE II method simultaneously considers the objectiveweights of the assessment criteria and the subjective pref-erences of the experts )us the proposed method identifiesthe top two net outranking flows as 0279 (bidder 1) and0212 (bidder 2) which are the winning bids )erefore theproposed extended PROMETHEE II approach is moresuitable for solving the contractor selection problem

43 Case Project 3 Case Project 3 involved the selection ofthe best contractor for a local Koji pottery industry pro-motion company in Chiayi Taiwan [42] In this local Kojipottery industry promotion case there were three biddersand the final evaluation committee consisted of four com-mittee members )e assessment criteria included thecontent of service proposal (C1) service management andcontrol of the project (C2) bidderrsquos bid price and compo-nents (C3) project ideas and feedback (C4) and brief andresponsive (C5) )e relative weights for the five assessment

Table 1 Actual evaluation data of software supplier

BidderCriteria

C1 C2 C3 C4 C5Bidder 1 62 238 231 156 72Bidder 2 60 230 230 150 63Bidder 3 80 263 263 175 85Bidder 4 67 215 223 148 60Bidder 5 88 260 270 170 85Bidder 6 82 245 263 167 80

Table 2 Software supplier selection by the traditional weightedarithmetic averaging method

BidderCriteria

Total score Actual winnerC1 C2 C3 C4 C5

Bidder 1 62 238 231 156 72 759Bidder 2 60 230 230 150 63 733Bidder 3 80 263 263 175 85 876 YesBidder 4 67 215 223 148 60 713Bidder 5 88 260 270 170 85 870Bidder 6 82 245 263 167 80 847


criteria were 030 025 015 020 and 010 respectivelyPerformance evaluation scores of each bidder are listed inTable 16

431 Solution by the Traditional Weighted Arithmetic Av-erage Approach Some information provided by Expert P3were missing or nonexistent data and thus only informa-tion provided by Experts P1 P2 and P4 were consideredAccording to the results of Table 16 equation (9) was used tocalculate the bidder scores for local Koji pottery industrypromotion contractor selection as shown in Table 17

432 Solution by the DEA Method From Table 16 ExpertsP1 P2 and P4 provided complete evaluation informationwhile Expert P3rsquos evaluation information was incomplete)e DEA method can only work with complete expert in-formation )erefore only the information provided byExperts P1 P2 and P4 were considered According to Ta-ble 16 the DEAP software was used to run the DEA CCRmodel and the local Koji pottery industry promotioncompany selection evaluation result is shown in Table 18

433 Solution by the Proposed Method )e proposed novelcontractor selection technique is not only able to handle theobjective weights of assessment criteria and subjectivepreferences of experts but can also handle missing ornonexistent contractor selection assessment process data byusing the average value of the complete information tocomplete the information According to Table 16 equations(1) and (2) were used to compute normalized decisionmatrix for the local Koji pottery industry promotioncompanies as listed in Table 19

According to Table 19 quations (3)ndash(5) were used tocalculate the preference function Pj(a b) and aggregatedpreference function for different local Koji pottery industrypromotion companies as shown in Table 20

According to Table 20 equations (6)ndash(8) were used tocalculate the entering flow leaving flow and net outrankingflow of different local Koji pottery industry promotioncompanies as shown in Table 21

434 Summary Case Project 3 involved the selection of thebest bidder as a local Koji pottery industry promotioncompany in Taiwan Table 22 shows the calculation results ofthe traditional weighted arithmetic averaging method theDEA method and the proposed method for Case Project 3

Table 3 Software supplier DEA evaluation results

DMU Output 1 Output 2 Output 3 Output 4 Output 5 Input DEA value () Actual winnerDMU 1 62 238 231 156 72 10 905DMU 2 60 230 230 150 63 10 875DMU 3 80 263 263 175 85 10 1000 YesDMU 4 67 215 223 148 60 10 847DMU 6 88 260 270 170 85 10 1000 YesDMU 6 82 245 263 167 80 10 978

Table 4 Normalized the decision matrix for different softwaresuppliers

BidderCriteria


Table 5 Aggregated preference function for different softwaresuppliers

Bidder Bidder1

Bidder2

Bidder3

Bidder4

Bidder5

Bidder6

Bidder 1 mdash 0144 0000 0302 0000 0000Bidder 2 0000 mdash 0000 0165 0000 0000Bidder 3 0618 0762 mdash 0902 0056 0192Bidder 4 0018 0025 0000 mdash 0000 0000Bidder 5 0635 0779 0073 0919 mdash 0202Bidder 6 0433 0577 0007 0717 0000 mdash

Table 6 Entering and leaving flows for different software suppliers

Bidder Entering flow Leaving flowBidder 1 0341 0089Bidder 2 0457 0033Bidder 3 0016 0506Bidder 4 0601 0009Bidder 5 0011 0522Bidder 6 0079 0347

Table 7 Net outranking flow values for different alternativesoftware suppliers

Bidder Net outranking flow RankBidder 1 minus0251 4Bidder 2 minus0424 5Bidder 3 0490 2Bidder 4 minus0593 6Bidder 5 0511 1Bidder 6 0268 3


Table 8 Results of different calculation methods for Case Project 1

BidderTraditional weighted

arithmetic averaging method DEA method )e proposed method

Total score Ranking DEA value () Ranking Net outranking flow RankingBidder 1 759 4 905 4 ndash0251 4Bidder 2 733 5 875 5 ndash0424 5Bidder 3 876 1 1000 1 0490 2Bidder 4 713 6 847 6 ndash0593 6Bidder 5 870 2 1000 1 0511 1Bidder 6 847 3 978 3 0268 3

Table 9 Actual evaluation data of security services companies

BidderCriteria

C1 C2 C3 C4 C5Bidder 1 180 205 202 175 78Bidder 2 130 220 210 170 80Bidder 3 183 197 188 168 78Bidder 4 125 185 173 172 72Bidder 5 127 213 201 161 77Bidder 6 174 201 191 170 80Bidder 7 169 199 189 170 80

Table 10 Security services company selection by the traditional weighted arithmetic average approach

BidderCriteria


Bidder 1 180 205 202 175 78 840 YesBidder 2 130 220 210 170 80 810Bidder 3 183 197 188 168 78 815Bidder 4 125 185 173 172 72 727Bidder 5 127 213 201 161 77 780Bidder 6 174 201 191 170 80 817 YesBidder 7 169 199 189 170 80 806

Table 11 Security services company DEA evaluation results

DMU Output 1 Output 2 Output 3 Output 4 Output 5 Input DEA value () Actual winnerDMU 1 180 205 202 175 78 10 1000 YesDMU 2 130 220 210 170 80 10 1000 YesDMU 3 183 197 188 168 78 10 1000 YesDMU 4 125 185 173 172 72 10 983DMU 5 127 213 201 161 77 10 969DMU 6 174 201 191 170 80 10 1000 YesDMU 7 169 199 189 170 80 10 1000 Yes

Table 12 Normalized the decision matrix of different security companies

BidderCriteria



Table 13 Aggregated preference function of different security companies

Bidder Bidder 1 Bidder 2 Bidder 3 Bidder 4 Bidder 5 Bidder 6 Bidder 7Bidder 1 mdash 0244 0252 0646 0402 0195 0240Bidder 2 0186 mdash 0367 0617 0287 0264 0292Bidder 3 0010 0183 mdash 0462 0306 0031 0048Bidder 4 0000 0029 0057 mdash 0157 0029 0029Bidder 5 0057 0000 0202 0459 mdash 0153 0181Bidder 6 0025 0152 0102 0505 0328 mdash 0045Bidder 7 0025 0134 0075 0460 0311 0000 mdash

Table 14 Entering leaving and net flows for different security companies

Bidder Entering flow Leaving flow Net outranking flowBidder 1 0051 0330 0279Bidder 2 0124 0336 0212Bidder 3 0176 0173 ndash0002Bidder 4 0525 0050 ndash0475Bidder 5 0299 0175 ndash0123Bidder 6 0112 0193 0081Bidder 7 0139 0167 0028




Total score Ranking DEA value () Ranking Net outranking flow RankingBidder 1 840 1 1000 1 0279 1Bidder 2 810 4 1000 1 0212 2Bidder 3 815 3 1000 1 ndash0002 5Bidder 4 727 7 983 6 ndash0475 7Bidder 5 780 6 969 7 ndash0123 6Bidder 6 817 2 1000 1 0081 3Bidder 7 806 5 1000 1 0028 4

Table 16 Performance evaluation scores of local Koji pottery industry promotion contractors

Criterion Weighting () Expert Bidder 1 Bidder 2 Bidder 3

C1 30

P1 25 24 25P2 24 25 25P3 24 23 24P4 24 25 24

C2 25

P1 23 23 22P2 22 22 21P3 lowast lowast lowast

P4 21 20 21

C3 15

P1 11 13 12P2 11 12 12P3 12 13 12P4 12 13 12

C4 20

P1 17 18 17P2 16 16 17P3 15 16 18P4 15 17 17

C5 10

P1 8 7 8P2 6 6 7P3 7 7 8P4 8 7 7

lowastindicates missing or nonexistent data


From Table 22 the DEA values of bidders 1 2 and 3 are100 which illustrates the high repetition rate problem ofthe DEAmethodMoreover neither the traditional weightedarithmetic averaging method nor the DEA method can

handle incomplete contractor selection process information)erefore the proposedmethod is more a general contractorselection technique and better suited for handling real-worldproblems

Table 18 DEA local Koji pottery industry promotion contractor evaluation result

Bidder Output 1 Output 2 Output 3 Output 4 Output 5 Input DEA value () Actual winnerBidder 1 243 220 113 160 73 10 1000 YesBidder 2 247 217 127 170 67 10 1000 YesBidder 3 247 213 120 170 73 10 1000 Yes

Table 19 Normalized the decision matrix for different promotion contractors

BidderCriteria

C1 C2 C3 C4 C5Bidder 1 0000 1000 0000 0000 0667Bidder 2 0000 0500 1000 0667 0000Bidder 3 1000 0000 0400 1000 1000

Table 20 Aggregated preference function for different promotion contractors

Bidder Bidder 1 Bidder 2 Bidder 3Bidder 1 mdash 0192 0250Bidder 2 0283 mdash 0215Bidder 3 0593 0467 mdash

Table 17 Local Koji pottery industry promotion contractor selection by the traditional weighted arithmetic averaging method

Bidder ExpertCriteria


Bidder 1P1 25 23 11 17 8 810P2 24 22 11 16 6P4 24 21 12 15 8

Bidder 2P1 24 23 13 18 7 827 YesP2 25 22 12 16 6P4 25 20 13 17 7

Bidder 3P1 25 22 12 17 8 823P2 25 21 12 17 7P4 24 21 12 17 7

Table 21 Entering leaving and net flows of different promotion contractors

Bidder Entering flow Leaving flow Net outranking flowBidder 1 0438 0221 minus0218Bidder 2 0329 0249 minus0080Bidder 3 0233 0530 0298




Total score Ranking DEA value () Ranking Net outranking flow RankingBidder 1 810 3 1000 1 minus0218 3Bidder 2 827 1 1000 1 minus0080 2Bidder 3 823 2 1000 1 0298 1


5 Conclusions

Contractor selection is a critical component of any com-panyrsquos pursuit of sustainable development in additionchoosing suitable contractors ensures that project man-agement plans are executed as expected However con-tractor selection includes qualitative and quantitativeassessment criteria at the same time which becomes acomplicated MCDM problem Traditional contractor se-lection methods are unable to handle nonexistent ormissing assessment criteria data nor consider the relativeimportance of criteria in the contractor selection assess-ment process In order to effectively address this problemthis study extended the PROMETHEE II method to pro-pose a novel contractor selection technique Moreoverthree numerical examples are applied to prove the cor-rectness and effectiveness of the proposed technique )esimulation results showed that the proposed extendedPROMETHEE II approach is a more general contractorselection technique for handling incomplete informationthan the traditional weighted arithmetic averaging methodand the DEA method

)e main advantages of the proposed extendedPROMETHEE II approach are as follows

(1) )e proposed approach can handle incomplete as-sessment criteria information

(2) )e proposed approach considers the relative im-portance of criteria

(3) )e proposed approach considers the subjectivepreferences of experts

(4) )e traditional weighted arithmetic average ap-proach and DEA method can be viewed as specialcases of the proposed approach

(5) )e proposed approach provides a more flexiblecontractor selection technique to support contractorassessment for selection

)is paper assumed that the experts have the sameweight and did not consider the objective weight of theevaluation data Further research studies can explore thesimultaneous considerations of subjective and objectiveweights of assessment criteria to handle different field de-cision-making issues

Data Availability

)e contractor selection data used to support the findings ofthis study are included within the article

Conflicts of Interest

)e author declares that there are no conflicts of interest

Acknowledgments

)e authors would like to thank the Ministry of Science andTechnology Taiwan for financially supporting this research

under Contract Nos MOST 109-2410-H-145-002 andMOST 110-2410-H-145-001

References

[1] J R San Cristobal ldquoContractor selection using multicriteriadecision-making methodsrdquo Journal of Construction Engi-neering and Management vol 138 no 6 pp 751ndash758 2012

[2] J B Yang H H Wang W C Wang and S M Ma ldquoUsingdata envelopment analysis to support best-value contractorselectionrdquo Journal of Civil Engineering and Managementvol 22 no 2 pp 199ndash209 2015

[3] C Akcay E Manisali and E Manisalı ldquoFuzzy decisionsupport model for the selection of contractor in constructionworksrdquo Revista de la construccion vol 17 no 2 pp 258ndash2662018

[4] M Hasnain M J )aheem and F Ullah ldquoBest value con-tractor selection in road construction projects ANP-baseddecision support systemrdquo International Journal of Civil En-gineering vol 16 no 6A pp 695ndash714 2018

[5] K H Chang ldquoA novel supplier selection method that inte-grates the intuitionistic fuzzy weighted averaging method anda soft set with imprecise datardquo Annals of Operations Researchvol 272 no 1-2 pp 139ndash157 2019

[6] G Gharedaghi and M Omidvari ldquoA pattern of contractorselection for oil and gas industries in a safety approach usingANP-DEMATEL in a Grey environmentrdquo InternationalJournal of Occupational Safety and Ergonomics vol 25 no 4pp 510ndash523 2019

[7] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash311999

[8] S Danjuma T Herawan M A Ismail H ChiromaA I Abubakar and A M Zeki ldquoA review on soft set-basedparameter reduction and decision makingrdquo IEEE Accessvol 5 pp 4671ndash4689 2017

[9] K-H Chang ldquoA more general risk assessment methodologyusing a soft set-based ranking techniquerdquo Soft Computingvol 18 no 1 pp 169ndash183 2014

[10] B Biswas S Bhattacharyya A Chakrabarti K N DeyJ Platos and V Snasel ldquoColonoscopy contrast-enhanced byintuitionistic fuzzy soft sets for polyp cancer localizationrdquoApplied Soft Computing vol 95 Article ID 106492 2020

[11] K-H Chang ldquoA novel reliability allocation approach usingthe OWA tree and soft setrdquo Annals of Operations Researchvol 244 no 1 pp 3ndash22 2016

[12] Z Tao H Chen L Zhou and J Liu ldquo2-Tuple linguistic softset and its application to group decision makingrdquo SoftComputing vol 19 no 5 pp 1201ndash1213 2015

[13] F Hao Z Pei D-S Park V Phonexay and H-S SeoldquoMobile cloud services recommendation a soft set-basedapproachrdquo Journal of Ambient Intelligence and HumanizedComputing vol 9 no 4 pp 1235ndash1243 2018

[14] B Paik and S K Mondal ldquoA distance-similarity method tosolve fuzzy sets and fuzzy soft sets based decision-makingproblemsrdquo Soft Computing vol 24 no 7 pp 5217ndash52292020

[15] M Abbas M Ali and S Romaguera ldquoGeneralized operationsin soft set theory via relaxed conditions on parametersrdquoFilomat vol 31 no 19 pp 5955ndash5964 2017

[16] K-H Chang ldquoEnhanced assessment of a supplier selectionproblem by integration of soft sets and hesitant fuzzy lin-guistic term setrdquo Proceedings of the Institution of Mechanical


EngineersmdashPart B Journal of Engineering Manufacturevol 229 no 9 pp 1635ndash1644 2015

[17] Z Kong J Zhao L Wang and J Zhang ldquoA new data fillingapproach based on probability analysis in incomplete softsetsrdquo Expert Systems with Applications vol 184 Article ID115358 2021

[18] M Akram M Shabir A Adeel and A N Al-Kenani ldquoAmultiattribute decision-making framework VIKOR methodwith complex spherical fuzzy N-soft setsrdquo MathematicalProblems in Engineering vol 2021 Article ID 1490807 2021

[19] T C Wen K H Chang and H H Lai ldquoIntegrating the 2-tuple linguistic representation and soft set to solve supplierselection problems with incomplete informationrdquo Engineer-ing Applications of Artificial Intelligence vol 87 Article ID103248 2020

[20] O Dalkilic ldquoA novel approach to soft set theory in decision-making under uncertaintyrdquo International Journal of Com-puter Mathematics vol 98 no 10 pp 11935ndash11945 2021

[21] J P Brans and P Vincke ldquoNote-A preference ranking or-ganisation methodrdquo Management Science vol 31 no 6pp 647ndash656 1985

[22] J M Brankovic M Markovic and D Nikolic ldquoComparativestudy of hydraulic structures alternatives using promethee IIcomplete ranking methodrdquo Water Resources Managementvol 32 no 10 pp 3457ndash3471 2018

[23] X J Tian X D Liu and L D Wang ldquoAn improvedPROMETHEE II method based on axiomatic fuzzy setsrdquoNeural Computing and Applications vol 25 no 7-8pp 1675ndash1683 2014

[24] F Samanlioglu and Z Ayag ldquoA fuzzy AHP-PROMETHEE IIapproach for evaluation of solar power plant location alter-natives in Turkeyrdquo Journal of Intelligent amp Fuzzy Systemsvol 33 no 2 pp 859ndash871 2017

[25] M Rabbani R Heidari and H Farrokhi-Asl ldquoA bi-objectivemixed-model assembly line sequencing problem consideringcustomer satisfaction and customer buying behaviourrdquo En-gineering Optimization vol 50 no 12 pp 2123ndash2142 2018

[26] D Jovanovic and D Cvetkovic ldquoMultiple decision makingcriteria in the implementation of renewable energy sourcesrdquoTehnicki Vjesnik-Technical Gazette vol 25 no 5 pp 1492ndash1496 2018

[27] W Zhang C Wang L Zhang et al ldquoEvaluation of theperformance of distributed and centralized biomass tech-nologies in rural Chinardquo Renewable Energy vol 125pp 445ndash455 2018

[28] D F D Silva J C S Silva L G O Silva L Ferreira andA T de Almeida ldquoSovereign credit risk assessment withmultiple criteria using an outranking methodrdquoMathematicalProblems in Engineering vol 2018 Article ID 856476411 pages 2018

[29] A d S Barbosa R A Shayani and M A G d Oliveira ldquoAmulti-criteria decision analysis method for regulatory eval-uation of electricity distribution service qualityrdquo UtilitiesPolicy vol 53 pp 38ndash48 2018

[30] G Markovic N Zdravkovic M Karakasic and M KolarevicldquoModified PROMETHEE approach for solving multi-criterialocation problems with complex criteria functionsrdquo TehnickiVjesnik-Technical Gazette vol 27 no 1 pp 12ndash19 2020

[31] L Z Tong Z M Pu K Chen and J J Yi ldquoSustainablemaintenance supplier performance evaluation based on anextend fuzzy PROMETHEE II approach in petrochemicalindustryrdquo Journal of Cleaner Production vol 273 Article ID122771 2020

[32] R Liu Y J Zhu Y Chen and H C Liu ldquoOccupational healthand safety risk assessment using an integrated TODIM-PROMETHEE model under linguistic spherical fuzzy envi-ronmentrdquo International Journal of Intelligent Systems vol 36no 11 pp 6814ndash6836 2021

[33] W Torbacki ldquoA hybrid MCDMmodel combining DANP andPROMETHEE II methods for the assessment of cybersecurityin industry 40rdquo Sustainability vol 13 no 16 p 8833 2021

[34] M S Roodposhti S Rahimi and M J Beglou ldquoPROM-ETHEE II and fuzzy AHP an enhanced GIS-based landslidesusceptibility mappingrdquo Natural Hazards vol 73 no 1pp 77ndash95 2014

[35] A Kessili and S Benmamar ldquoPrioritizing sewer rehabilitationprojects using AHP-PROMETHEE II ranking methodrdquoWater Science and Technology vol 73 no 2 pp 283ndash2912016

[36] S R Maity and S Chakraborty ldquoTool steel material selectionusing PROMETHEE II methodrdquo International Journal ofAdvanced Manufacturing Technology vol 78 no 9-12pp 1537ndash1547 2015

[37] M I Ali M Shabir and F Feng ldquoRepresentation of graphsbased on neighborhoods and soft setsrdquo International Journalof Machine Learning and Cybernetics vol 8 no 5pp 1525ndash1535 2017

[38] P K Maji R Biswas and A R Roy ldquoSoft set theoryrdquoComputers and Mathematics with Applications vol 45 no 4-5 pp 555ndash562 2003

[39] K H Chang Y C Chang K Chain and H Y ChungldquoIntegrating soft set theory and fuzzy linguistic model toevaluate the performance of training simulation systemsrdquoPLoS One vol 11 no 9 Article ID e0162092 2016

[40] Z Xu ldquoA note on linguistic hybrid arithmetic averagingoperator in multiple attribute group decision making withlinguistic informationrdquo Group Decision and Negotiationvol 15 no 6 pp 593ndash604 2006

[41] J M Merigo ldquoA unified model between the weighted averageand the induced OWA operatorrdquo Expert Systems with Ap-plications vol 38 no 9 pp 11560ndash11572 2011

[42] K H Chang ldquoA general soft set and the OWA approach for asupplier selection problem under incomplete informationrdquoICIC Express Letters Part B Applications vol 4 no 2pp 395ndash400 2013


results Fortunately the soft set method is able to handleincomplete assessment criteria information )e soft setmethod was first proposed by Molodtsov [7] to handle theuncertainty of data or traditional mathematical tools unableto handle fuzzy information in the process of informationfusion and following the development of soft sets numerousstudies used the soft set method to handle different decision-making problems [7ndash20]

Contractors play a key role in successful operations andoverall project performance in addition choosing the rightcontractors for a particular project is a major challenge in thesupply chain However traditional contractor selectionmethods do not consider information regarding the relativeimportance of criteria )e preference ranking organizationmethod for enrichment evaluation (PROMETHEE) methodis one of the most commonly used techniques to solveMCDM issues)e advantages of the PROMETHEEmethodare that the relative importance of criteria is consideredPROMETHEE method was first introduced by Brans andVincke [21] applied as an outranking relation techniquebetween pairs of alternatives to solve MCDM problemsBecause the PROMETHEE methodrsquos calculation is simpleand it is easy to operate numerous studies have used thismethod to handle decision-making problems in variousfields For example Brankovic et al [22] used PROMETHEEand the simplified elimination and choice translating reality(ELECTRE) approach to determine the criterion weight forselecting optimal alternative hydraulic structure solutionsTian et al [23] proposed an improved PROMETHEE IImethod based on axiomatic fuzzy set theory which si-multaneously considered the subjective preferences of ex-perts and the objective weights of assessment criteria )eimproved PROMETHEE II method considers ranking aswell as the degree of credibility of the raw data )ePROMETHEEmethod has been used in many fields such asalternative locations for solar power plants [24] the as-sembly line sequencing problem [25] renewable energysource assessment [26 27] credit risk assessment [28]electricity distribution utility performance assessment [29]logistics warehouse location selection [30] petrochemicalindustry [31] occupational health and safety [32] andcybersecurity of Industry 40 [33] However the typicalPROMETHEE method cannot handle the incomplete as-sessment criteria information

Recently Yang et al [2] introduced a DEA-based ap-proach to support the selection of the best value contractorHowever this approach would cause a high repetition rateproblem when DEA values are 100 Moreover this ap-proach cannot handle cases when the needed expert data ismissing or nonexistent during the information assessmentprocess In order to effectively resolve the above contractorselection issues this study proposes a flexible PROMETHEEII method to deal with the contractor selection issue whichsimultaneously includes complete and incomplete infor-mation To verify the effectiveness of the proposed flexiblePROMETHEE II approach three contractor selection caseswere adopted and the simulation results were comparedwith the traditional weighted arithmetic averaging methodand the DEA method [2]

)e major contributions of this paper include the fol-lowing advantages (1) the proposed approach can handleincomplete assessment criteria information (2) the pro-posed approach considers the relative importance of criteria(3) the proposed approach considers the subjective pref-erences of experts (4) the traditional weighted arithmeticaverage approach and DEA method can be viewed as specialcases of the proposed approach and (5) the proposed ap-proach provides a more flexible contractor selection tech-nique to support contractor assessment for selection

)e remainder of this paper is arranged as followsSection 2 reviews related research on the PROMETHEE IImethod and soft set method Section 3 introduces theproposed novel contractor selection approach for solvingbest-value tendering selection problems )ree contractorselection case projects are adopted and the comparison withother related methods are discussed in Section 4 Finally theconclusion is given in Section 5

2 Preliminaries

)is section presents some fundamental definitions andconcepts related to the PROMETHEE II method and soft set

21 PROMETHEE II Method Brans and Vincke [21] pro-posed the PROMETHEE II approach that based on thepairwise comparison of alternatives for each criterion tosolve MCDM problems Two types of information are re-quired for the PROMETHEE II approach (1) the relativeimportance of the criteria and (2) decision-makerrsquos pref-erence function for comparing alternative contributions[34 35]

)e PROMETHEE II method consists of five steps [36]

Step 1 normalize the decision matrix

Rij xij minus min xij1113872 11138731113960 1113961

max xij1113872 1113873 minus min xij1113872 1113873(i 1 2 n j 1 2 m)

(1)

where xij is the performance index of the ith alternativeof the jth criterionFor nonbeneficial criteria equation (1) can be rewrittenas follows

Rij max xij1113872 1113873 minus xij1113960 1113961

max xij1113872 1113873 minus min xij1113872 1113873i 1 2 n j 1 2 m

(2)

Step 2 calculate the preference function Pj(a b) asfollows

Pj(a b) 0 if Raj leRbj (3)

Pj(a b) Raj minus Rbj1113872 1113873 if Raj gtRbj (4)

Step 3 calculate the aggregated preference functionπ(a b) as follows


π(a b) 1113944m

j1wjPj(a b) (5)


φminus(a)

1m minus 1

1113944

m

b1π(b a) (6)

φ+(a)

1m minus 1

1113944

m

b1π(a b) (7)


(a) (8)













⎧⎪⎨

⎪⎩















WWA a1 a2 an( 1113857 1113944 ni1wiai (9)















Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Step 7

Step 8

Step 9

















BidderCriteria



BidderCriteria














BidderCriteria



Bidder Bidder1

Bidder2

Bidder3

Bidder4

Bidder5

Bidder6












BidderCriteria



BidderCriteria






BidderCriteria













C1 30

P1 25 24 25P2 24 25 25P3 24 23 24P4 24 25 24

C2 25


P4 21 20 21

C3 15

P1 11 13 12P2 11 12 12P3 12 13 12P4 12 13 12

C4 20

P1 17 18 17P2 16 16 17P3 15 16 18P4 15 17 17

C5 10

P1 8 7 8P2 6 6 7P3 7 7 8P4 8 7 7








BidderCriteria







Bidder 1P1 25 23 11 17 8 810P2 24 22 11 16 6P4 24 21 12 15 8

Bidder 2P1 24 23 13 18 7 827 YesP2 25 22 12 16 6P4 25 20 13 17 7

Bidder 3P1 25 22 12 17 8 823P2 25 21 12 17 7P4 24 21 12 17 7








5 Conclusions









Data Availability




Acknowledgments



References














































π(a b) 1113944m

j1wjPj(a b) (5)


φminus(a)

1m minus 1

1113944

m

b1π(b a) (6)

φ+(a)

1m minus 1

1113944

m

b1π(a b) (7)


(a) (8)













⎧⎪⎨

⎪⎩















WWA a1 a2 an( 1113857 1113944 ni1wiai (9)















Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Step 7

Step 8

Step 9

















BidderCriteria



BidderCriteria














BidderCriteria



Bidder Bidder1

Bidder2

Bidder3

Bidder4

Bidder5

Bidder6












BidderCriteria



BidderCriteria






BidderCriteria













C1 30

P1 25 24 25P2 24 25 25P3 24 23 24P4 24 25 24

C2 25


P4 21 20 21

C3 15

P1 11 13 12P2 11 12 12P3 12 13 12P4 12 13 12

C4 20

P1 17 18 17P2 16 16 17P3 15 16 18P4 15 17 17

C5 10

P1 8 7 8P2 6 6 7P3 7 7 8P4 8 7 7








BidderCriteria







Bidder 1P1 25 23 11 17 8 810P2 24 22 11 16 6P4 24 21 12 15 8

Bidder 2P1 24 23 13 18 7 827 YesP2 25 22 12 16 6P4 25 20 13 17 7

Bidder 3P1 25 22 12 17 8 823P2 25 21 12 17 7P4 24 21 12 17 7








5 Conclusions









Data Availability




Acknowledgments



References






















































WWA a1 a2 an( 1113857 1113944 ni1wiai (9)















Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Step 7

Step 8

Step 9

















BidderCriteria



BidderCriteria














BidderCriteria



Bidder Bidder1

Bidder2

Bidder3

Bidder4

Bidder5

Bidder6












BidderCriteria



BidderCriteria






BidderCriteria













C1 30

P1 25 24 25P2 24 25 25P3 24 23 24P4 24 25 24

C2 25


P4 21 20 21

C3 15

P1 11 13 12P2 11 12 12P3 12 13 12P4 12 13 12

C4 20

P1 17 18 17P2 16 16 17P3 15 16 18P4 15 17 17

C5 10

P1 8 7 8P2 6 6 7P3 7 7 8P4 8 7 7








BidderCriteria







Bidder 1P1 25 23 11 17 8 810P2 24 22 11 16 6P4 24 21 12 15 8

Bidder 2P1 24 23 13 18 7 827 YesP2 25 22 12 16 6P4 25 20 13 17 7

Bidder 3P1 25 22 12 17 8 823P2 25 21 12 17 7P4 24 21 12 17 7








5 Conclusions









Data Availability




Acknowledgments



References




























































BidderCriteria



BidderCriteria














BidderCriteria



Bidder Bidder1

Bidder2

Bidder3

Bidder4

Bidder5

Bidder6












BidderCriteria



BidderCriteria






BidderCriteria













C1 30

P1 25 24 25P2 24 25 25P3 24 23 24P4 24 25 24

C2 25


P4 21 20 21

C3 15

P1 11 13 12P2 11 12 12P3 12 13 12P4 12 13 12

C4 20

P1 17 18 17P2 16 16 17P3 15 16 18P4 15 17 17

C5 10

P1 8 7 8P2 6 6 7P3 7 7 8P4 8 7 7








BidderCriteria







Bidder 1P1 25 23 11 17 8 810P2 24 22 11 16 6P4 24 21 12 15 8

Bidder 2P1 24 23 13 18 7 827 YesP2 25 22 12 16 6P4 25 20 13 17 7

Bidder 3P1 25 22 12 17 8 823P2 25 21 12 17 7P4 24 21 12 17 7








5 Conclusions









Data Availability




Acknowledgments



References
























































BidderCriteria



Bidder Bidder1

Bidder2

Bidder3

Bidder4

Bidder5

Bidder6












BidderCriteria



BidderCriteria






BidderCriteria













C1 30

P1 25 24 25P2 24 25 25P3 24 23 24P4 24 25 24

C2 25


P4 21 20 21

C3 15

P1 11 13 12P2 11 12 12P3 12 13 12P4 12 13 12

C4 20

P1 17 18 17P2 16 16 17P3 15 16 18P4 15 17 17

C5 10

P1 8 7 8P2 6 6 7P3 7 7 8P4 8 7 7








BidderCriteria







Bidder 1P1 25 23 11 17 8 810P2 24 22 11 16 6P4 24 21 12 15 8

Bidder 2P1 24 23 13 18 7 827 YesP2 25 22 12 16 6P4 25 20 13 17 7

Bidder 3P1 25 22 12 17 8 823P2 25 21 12 17 7P4 24 21 12 17 7








5 Conclusions









Data Availability




Acknowledgments



References



















































BidderCriteria



BidderCriteria






BidderCriteria













C1 30

P1 25 24 25P2 24 25 25P3 24 23 24P4 24 25 24

C2 25


P4 21 20 21

C3 15

P1 11 13 12P2 11 12 12P3 12 13 12P4 12 13 12

C4 20

P1 17 18 17P2 16 16 17P3 15 16 18P4 15 17 17

C5 10

P1 8 7 8P2 6 6 7P3 7 7 8P4 8 7 7








BidderCriteria







Bidder 1P1 25 23 11 17 8 810P2 24 22 11 16 6P4 24 21 12 15 8

Bidder 2P1 24 23 13 18 7 827 YesP2 25 22 12 16 6P4 25 20 13 17 7

Bidder 3P1 25 22 12 17 8 823P2 25 21 12 17 7P4 24 21 12 17 7








5 Conclusions









Data Availability




Acknowledgments



References
























































C1 30

P1 25 24 25P2 24 25 25P3 24 23 24P4 24 25 24

C2 25


P4 21 20 21

C3 15

P1 11 13 12P2 11 12 12P3 12 13 12P4 12 13 12

C4 20

P1 17 18 17P2 16 16 17P3 15 16 18P4 15 17 17

C5 10

P1 8 7 8P2 6 6 7P3 7 7 8P4 8 7 7








BidderCriteria







Bidder 1P1 25 23 11 17 8 810P2 24 22 11 16 6P4 24 21 12 15 8

Bidder 2P1 24 23 13 18 7 827 YesP2 25 22 12 16 6P4 25 20 13 17 7

Bidder 3P1 25 22 12 17 8 823P2 25 21 12 17 7P4 24 21 12 17 7








5 Conclusions









Data Availability




Acknowledgments



References



















































BidderCriteria







Bidder 1P1 25 23 11 17 8 810P2 24 22 11 16 6P4 24 21 12 15 8

Bidder 2P1 24 23 13 18 7 827 YesP2 25 22 12 16 6P4 25 20 13 17 7

Bidder 3P1 25 22 12 17 8 823P2 25 21 12 17 7P4 24 21 12 17 7








5 Conclusions









Data Availability




Acknowledgments



References














































5 Conclusions









Data Availability




Acknowledgments



References














































A Novel Contractor Selection Technique Using the Extended ...

Documents

Transcript of A Novel Contractor Selection Technique Using the Extended ...