# A Novel Contractor Selection Technique Using the Extended PROMETHEE II Method

Authors: Kuei-Hu Chang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

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


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## Body

## 1. Introduction

Contractor selection includes multiple performance assessment criteria and is a multicriteria decision-making (MCDM) issue. Choosing suitable contractors directly affects the competitive advantage of products and the sustainable development of enterprises. Therefore, contractor selection is a critical issue in the supply chain and has received considerable research attention. Many authors have used different computation methods to address contractor selection problems. For example, San Cristobal [1] combined VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR), techniques for order preference by similarity to ideal solution (TOPSIS), and the analytical hierarchy process (AHP) methods for selecting suitable contractors for the “La Braguía” 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 value contractors in single-input and multiple-output manners. Akcay et al. [3] used the concept of fuzzy logic to propose a fuzzy decision support model for contractor selection of Turkish construction projects. Hasnain et al. [4] proposed an analytical network process (ANP) based decision support system to solve the most valuable contractor selection problems in road construction projects. Chang [5] integrated the soft set and intuitionistic fuzzy weighted average methods to select the best supplier under an incomplete information environment. Gharedaghi and Omidvari [6] integrated the analytical network processing, decision-making trial and evaluation laboratory (DEMATEL), and the grey system theory to select suitable safety contractors for oil and gas industries.The contractor selection assessment process may encounter instances of missing or nonexistent assessment criteria data. Incomplete attribute value information increases the difficulty of assessing contractors. In order to handle incomplete information, the traditional contractor selection method directly deletes incomplete attribute value information. Deleting information, however, results in a reduction in available information and distorts evaluation results. Fortunately, the soft set method is able to handle incomplete assessment criteria information. The soft set method was first proposed by Molodtsov [7] to handle the uncertainty of data or traditional mathematical tools unable to handle fuzzy information in the process of information fusion, and following the development of soft sets, numerous studies used the soft set method to handle different decision-making problems [7–20].Contractors play a key role in successful operations and overall project performance; in addition, choosing the right contractors for a particular project is a major challenge in the supply chain. However, traditional contractor selection methods do not consider information regarding the relative importance of criteria. The preference ranking organization method for enrichment evaluation (PROMETHEE) method is one of the most commonly used techniques to solve MCDM issues. The advantages of the PROMETHEE method are that the relative importance of criteria is considered. PROMETHEE method was first introduced by Brans and Vincke [21], applied as an outranking relation technique between pairs of alternatives to solve MCDM problems. Because the PROMETHEE method’s calculation is simple and it is easy to operate, numerous studies have used this method to handle decision-making problems in various fields. For example, Brankovic et al. [22] used PROMETHEE and the simplified elimination and choice translating reality (ELECTRE) approach to determine the criterion weight for selecting optimal alternative hydraulic structure solutions. Tian et al. [23] proposed an improved PROMETHEE II method based on axiomatic fuzzy set theory, which simultaneously considered the subjective preferences of experts and the objective weights of assessment criteria. The improved PROMETHEE II method considers ranking, as well as the degree of credibility of the raw data. The PROMETHEE method has been used in many fields, such as alternative locations for solar power plants [24], the assembly line sequencing problem [25], renewable energy source assessment [26, 27], credit risk assessment [28], electricity distribution utility performance assessment [29], logistics warehouse location selection [30], petrochemical industry [31], occupational health and safety [32], and cybersecurity of Industry 4.0 [33]. However, the typical PROMETHEE method cannot handle the incomplete assessment criteria information.Recently, Yang et al. [2] introduced a DEA-based approach to support the selection of the best value contractor. However, this approach would cause a high repetition rate problem when DEA values are 100%. Moreover, this approach cannot handle cases when the needed expert data is missing or nonexistent during the information assessment process. In order to effectively resolve the above contractor selection issues, this study proposes a flexible PROMETHEE II method to deal with the contractor selection issue, which simultaneously includes complete and incomplete information. To verify the effectiveness of the proposed flexible PROMETHEE II approach, three contractor selection cases were adopted, and the simulation results were compared with the traditional weighted arithmetic averaging method and the DEA method [2].The major contributions of this paper include the following advantages: (1) the proposed approach can handle incomplete assessment criteria information; (2) the proposed approach considers the relative importance of criteria; (3) the proposed approach considers the subjective preferences of experts; (4) the traditional weighted arithmetic average approach and DEA method can be viewed as special cases of the proposed approach; and (5) the proposed approach provides a more flexible contractor selection technique to support contractor assessment for selection.The remainder of this paper is arranged as follows. Section2 reviews related research on the PROMETHEE II method and soft set method. Section 3 introduces the proposed novel contractor selection approach for solving best-value tendering selection problems. Three contractor selection case projects are adopted, and the comparison with other related methods are discussed in Section 4. Finally, the conclusion is given in Section 5.


## 2. Preliminaries

This section presents some fundamental definitions and concepts related to the PROMETHEE II method and soft set.


### 2.1. PROMETHEE II Method

Brans and Vincke [21] proposed the PROMETHEE II approach that based on the pairwise comparison of alternatives for each criterion to solve MCDM problems. Two types of information are required for the PROMETHEE II approach: (1) the relative importance of the criteria and (2) decision-maker’s preference function for comparing alternative contributions [34, 35].The PROMETHEE II method consists of five steps [36]:Step 1: normalize the decision matrix(1)Rij=xij−minxijmaxxij−minxiji=1,2,…n;j=1,2,…,m,wherexij is the performance index of the ith alternative of the jth criterion.For nonbeneficial criteria, equation (1) can be rewritten as follows:(2)Rij=maxxij−xijmaxxij−minxiji=1,2,…n;j=1,2,…,m.Step 2: calculate the preference function,Pja,b as follows:(3)Pja,b=0ifRaj≤Rbj,(4)Pja,b=Raj−RbjifRaj>Rbj.Step 3: calculate the aggregated preference function,πa,b as follows:(5)πa,b=∑j=1mwjPja,b,wherewj is the weight of the jth criterion.Step 4: calculate the entering flow, leaving flow, and net flow for each alternative(6)φ−a=1m−1∑b=1mπb,a,(7)φ+a=1m−1∑b=1mπa,b,(8)φa=φ+a−φ−a,whereφ−a, φ+a, and φa represent the entering flow, leaving flow, and net flow for each alternative, respectively.Step 5: determine the ranking of all the possible alternatives.All possible alternatives are ranked according to the net flow valueφa, with a higher net flow value φa representing a better alternative.


### 2.2. Soft Set

The soft set method is a novel mathematical method developed by Molodtsov [7] to handle the related issues of uncertain and ambiguous data, and the method is explained as follows: assume U indicates an initial universe and E be the parameters set related to the objects in U. The set P(U) be the power set of U and A⊆E.Definition 1.
[7, 37]: A pair (F, A) is called a soft set (on U), where F is a mapping given by F:A⟶PU.
In other words, the soft set overU is a parameterized family of universe U subsets.Definition 2.
[7, 38]: Both two soft sets (F, X) and (G, Y) are in a common universe U, and (F, X) is the soft subset of (G, Y), represented as F,X⊂˜G,Y, if(i)
X ⊆ Y and(ii)
∀e ∈ X, F(e) ⊆ G(e).Definition 3.
[38, 39]: Both two soft sets (F, X) and (G, Y) are in a common universe U, where the union of (F, X) and (G, Y) is represented as (H, Z), and the following conditions are satisfied:(i)
Z=X∪Y and(ii)
∀e ∈ Z, He=Fe,ife∈X−Y,Ge,ife∈Y−X,Fe∪Ge,ife∈X∩Y..Definition 4.
[38, 39]: Both two soft sets (F, X) and (G, Y) are in a common universe U, where the intersection of (F, X) and (G, Y) is represented as (H, Z), and the following conditions are satisfied:(i)
Z=X∩Y and(ii)
∀e ∈ Z, He=FeorGe.


## 2.1. PROMETHEE II Method

Brans and Vincke [21] proposed the PROMETHEE II approach that based on the pairwise comparison of alternatives for each criterion to solve MCDM problems. Two types of information are required for the PROMETHEE II approach: (1) the relative importance of the criteria and (2) decision-maker’s preference function for comparing alternative contributions [34, 35].The PROMETHEE II method consists of five steps [36]:Step 1: normalize the decision matrix(1)Rij=xij−minxijmaxxij−minxiji=1,2,…n;j=1,2,…,m,wherexij is the performance index of the ith alternative of the jth criterion.For nonbeneficial criteria, equation (1) can be rewritten as follows:(2)Rij=maxxij−xijmaxxij−minxiji=1,2,…n;j=1,2,…,m.Step 2: calculate the preference function,Pja,b as follows:(3)Pja,b=0ifRaj≤Rbj,(4)Pja,b=Raj−RbjifRaj>Rbj.Step 3: calculate the aggregated preference function,πa,b as follows:(5)πa,b=∑j=1mwjPja,b,wherewj is the weight of the jth criterion.Step 4: calculate the entering flow, leaving flow, and net flow for each alternative(6)φ−a=1m−1∑b=1mπb,a,(7)φ+a=1m−1∑b=1mπa,b,(8)φa=φ+a−φ−a,whereφ−a, φ+a, and φa represent the entering flow, leaving flow, and net flow for each alternative, respectively.Step 5: determine the ranking of all the possible alternatives.All possible alternatives are ranked according to the net flow valueφa, with a higher net flow value φa representing a better alternative.


## 2.2. Soft Set

The soft set method is a novel mathematical method developed by Molodtsov [7] to handle the related issues of uncertain and ambiguous data, and the method is explained as follows: assume U indicates an initial universe and E be the parameters set related to the objects in U. The set P(U) be the power set of U and A⊆E.Definition 1.
[7, 37]: A pair (F, A) is called a soft set (on U), where F is a mapping given by F:A⟶PU.
In other words, the soft set overU is a parameterized family of universe U subsets.Definition 2.
[7, 38]: Both two soft sets (F, X) and (G, Y) are in a common universe U, and (F, X) is the soft subset of (G, Y), represented as F,X⊂˜G,Y, if(i)
X ⊆ Y and(ii)
∀e ∈ X, F(e) ⊆ G(e).Definition 3.
[38, 39]: Both two soft sets (F, X) and (G, Y) are in a common universe U, where the union of (F, X) and (G, Y) is represented as (H, Z), and the following conditions are satisfied:(i)
Z=X∪Y and(ii)
∀e ∈ Z, He=Fe,ife∈X−Y,Ge,ife∈Y−X,Fe∪Ge,ife∈X∩Y..Definition 4.
[38, 39]: Both two soft sets (F, X) and (G, Y) are in a common universe U, where the intersection of (F, X) and (G, Y) is represented as (H, Z), and the following conditions are satisfied:(i)
Z=X∩Y and(ii)
∀e ∈ Z, He=FeorGe.


## 3. The Proposed Novel Contractor Selection Method

Government procurement in Taiwan involves two principles for contractor selection: the best value bid and the lowest bid. The assessment criteria for the best value bid simultaneously include qualitative and quantitative data and present a complicated MCDM problem. However, the traditional contractor selection approach does not consider the relative importance of criteria or subjective preferences of experts. Moreover, instances may occur in the contractor selection assessment process in which assessment criteria data are missing or nonexistent. This will make the contractor selection evaluation more complicated and difficult. In order to effectively address this problem, this paper proposes a flexible PROMETHEE II method to deal with contractor selection problems, which simultaneously includes complete and incomplete information. This study uses the weighted arithmetic averaging method to fill in incomplete assessment criteria score data. Furthermore, this study applies the PROMETHEE II method to handle information about the relative importance of criteria and subjective preferences of experts. This study integrates the PROMETHEE II method and soft set method to select suitable contractors. The results are more suitably and flexibly reflect the actual situation. The implementation steps of the proposed novel contractor selection approach are as follows (depicted in Figure1):Step 1: confirm the bidding document assessment criteria.Determine the content of assessment criteria according to the requirements of the actual bidding documents.Step 2: determine the relative weights of the assessment criteria.etermine the relative weights of the assessment criteria according to the importance level of the assessment criteria.Step 3: determine the scores of the assessment criteria for each bidder.Each committee member determines the assessment criteria scores for each bidder.Step 4: fill in incomplete assessment criteria score data.If the assessment criteria scores are incomplete, use the weighted arithmetic average approach to fill in incomplete assessment criteria score data.Step 5: normalize the decision matrix.Use equations (1) and (2) to compute the normalized decision matrix.Step 6: compute the preference functionPja,b.Use equations (3) and (4) to compute the preference function Pja,b.Step 7: compute the aggregated preference functionπa,b.Use equation (5) to compute the aggregated preference function πa,b.Step 8: compute the entering flow and leaving flow for each alternative.Use equations (6)–(7) to compute the entering flow and leaving flow for each alternative.Step 9: identify the ranking of all possible alternatives.Use equation (8) to compute the net flow values for each alternative. Rank the net flow φa from highest to lowest, with a higher net flow value φa representing a better bidder.Figure 1
Flowchart of the proposed novel contractor selection method.


## 4. Illustrative Example

### 4.1. Case Project 1

This section presents an illustrative example of software supplier selection adapted from Yang et al. [2] who proved the effectiveness of the proposed novel contractor selection approach. The case company must select the optimal software supplier in order to develop and plan the management information system. There were six bidders, and a bid opening committee that consisted of four committee members in this software supplier case. The assessment criteria included the comprehension degree for the project (C1), system design and planning (C2), coding ability of a system (C3), management ability of project (C4), and past achievements and experience (C5). The relative weights of the five evaluation criteria were 0.10, 0.30, 0.30, 0.20, and 0.10, respectively. The five evaluation criteria were then used by the final bid opening committee to determine the most suitable software supplier. The five assessment criteria scores for each bidder are listed in Table 1.Table 1
Actual evaluation data of software supplier.
BidderCriteriaC1C2C3C4C5Bidder 16.223.823.115.67.2Bidder 26.023.023.015.06.3Bidder 38.026.326.317.58.5Bidder 46.721.522.314.86.0Bidder 58.826.027.017.08.5Bidder 68.224.526.316.78.0


#### 4.1.1. Solution by the Traditional Weighted Arithmetic Averaging Approach

The traditional weighted arithmetic averaging approach is one of the simplest and most commonly used aggregation operators, and it can be explained as follows.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 and ∑i=1nwi=1,wi∈0,1 as follows:(9)WWAa1,a2,…,an=∑i=1nwiai,where ai is the argument variable.
According to the results of Table1, equation (9) was used to calculate the bidder scores for software supplier selection, as shown in Table 2.Table 2
Software supplier selection by the traditional weighted arithmetic averaging method.
BidderCriteriaTotal scoreActual winnerC1C2C3C4C5Bidder 16.223.823.115.67.275.9Bidder 26.023.023.015.06.373.3Bidder 38.026.326.317.58.587.6YesBidder 46.721.522.314.86.071.3Bidder 58.826.027.017.08.587.0Bidder 68.224.526.316.78.084.7


#### 4.1.2. Solution by the DEA Method

Yang et al. [2] proposed a DEA model with single input and multiple outputs to solve the best-value contractor selection problem. They used the decision-making unit (DMU) as a bidder, and the DEA input values were equal to 1. According to Table 1, the DEAP software was used to run the DEA CCR model, and the software supplier evaluation results as listed in Table 3, where a bidder’s DEA value of 100% indicates that the bidder has the highest comparative efficiency.Table 3
Software supplier DEA evaluation results.
DMUOutput 1Output 2Output 3Output 4Output 5InputDEA value (%)Actual winnerDMU 16.223.823.115.67.21.090.5DMU 26.023.023.015.06.31.087.5DMU 38.026.326.317.58.51.0100.0YesDMU 46.721.522.314.86.01.084.7DMU 68.826.027.017.08.51.0100.0YesDMU 68.224.526.316.78.01.097.8


#### 4.1.3. Solution by the Proposed Novel Contractor Selection Approach

The proposed novel contractor selection approach integrates the PROMETHEE II method and the soft set approach to select suitable contractors. In the software supplier selection example, the five assessment criteria scores for six bidders in the final bid opening committee are as listed in Table1 (Steps 1 ∼ 4). The following procedure describes the remaining steps of the proposed method:Step 5: normalize the decision matrix.According to Table1, equations (1) and (2) were used to calculate the normalized decision matrix for different software suppliers, as listed in Table 4.Step 6: compute the preference functionPja,b.Step 7: compute the aggregated preference functionπa,b.According to Table4, equations (3)–(5) were used to compute the preference function (Pja,b) and aggregated preference function (πa,b) for different software suppliers, as listed in Table 5.Step 8: compute the entering flow and leaving flow for each alternative.According to Table5, equations (6) and (7) were used to calculate the entering and leaving flows for different software suppliers, as shown in Table 6.Step 9: determine the ranking of all possible alternatives.Table 4
Normalized the decision matrix for different software suppliers.
BidderCriteriaC1C2C3C4C5Bidder 10.0710.4790.1700.2960.480Bidder 20.0000.3130.1490.0740.120Bidder 30.7141.0000.8511.0001.000Bidder 40.2500.0000.0000.0000.000Bidder 51.0000.9381.0000.8151.000Bidder 60.7860.6250.8510.7040.800Table 5
Aggregated preference function for different software suppliers.
BidderBidder 1Bidder 2Bidder 3Bidder 4Bidder 5Bidder 6Bidder 1—0.1440.0000.3020.0000.000Bidder 20.000—0.0000.1650.0000.000Bidder 30.6180.762—0.9020.0560.192Bidder 40.0180.0250.000—0.0000.000Bidder 50.6350.7790.0730.919—0.202Bidder 60.4330.5770.0070.7170.000—Table 6
Entering and leaving flows for different software suppliers.
BidderEntering flowLeaving flowBidder 10.3410.089Bidder 20.4570.033Bidder 30.0160.506Bidder 40.6010.009Bidder 50.0110.522Bidder 60.0790.347According to Table6, equation (8) was used to calculate the net flow for different software suppliers. The net flow values φa was ranked from highest to lowest, with a higher net flow value φa representing a better software supplier, as listed in Table 7.Table 7
Net outranking flow values for different alternative software suppliers.
BidderNet outranking flowRankBidder 1−0.2514Bidder 2−0.4245Bidder 30.4902Bidder 4−0.5936Bidder 50.5111Bidder 60.2683


#### 4.1.4. Summary

Case Project 1 involves selecting a software supplier to develop a management information system for the case company. The input data for each bidder was listed in Table1. Table 8 lists the different calculation results of the traditional weighted arithmetic averaging method, the DEA method, and the proposed method for Case Project 1.Table 8
Results of different calculation methods for Case Project 1.
BidderTraditional weighted arithmetic averaging methodDEA methodThe proposed methodTotal scoreRankingDEA value (%)RankingNet outranking flowRankingBidder 175.9490.54–0.2514Bidder 273.3587.55–0.4245Bidder 387.61100.010.4902Bidder 471.3684.76–0.5936Bidder 587.02100.010.5111Bidder 684.7397.830.2683From Table8, the DEA values for bidders 3 and 5 were 100%. The proposed extended PROMETHEE II method can solve this problem of duplicate DEA value. In the proposed method, the net outranking flow of bidders 3 and 5 were 0.490 and 0.511, respectively. Therefore, the proposed method can better sort bidder scores.


### 4.2. Case Project 2

Case Project 2 involves the selection of two security companies to provide security services for a science park [2]. There were seven bidders competing for two awards in this case. The assessment criteria included company organization (C1), planning feasibility (C2), professional capability (C3), price (C4), and presentation and question response (C5). The relative weights for the five assessment criteria were 0.20, 0.25, 0.25, 0.20, and 0.10, respectively, and the final evaluation committee selected the best two security services companies based on the five assessment criteria. The five assessment criteria scores for each security services company are listed in Table 9.Table 9
Actual evaluation data of security services companies.
BidderCriteriaC1C2C3C4C5Bidder 118.020.520.217.57.8Bidder 213.022.021.017.08.0Bidder 318.319.718.816.87.8Bidder 412.518.517.317.27.2Bidder 512.721.320.116.17.7Bidder 617.420.119.117.08.0Bidder 716.919.918.917.08.0


#### 4.2.1. Solution by the Traditional Weighted Arithmetic Average Approach

Based on the results of Table9, equation (9) was used to calculate the bidder scores for security services company selection, as shown in Table 10.Table 10
Security services company selection by the traditional weighted arithmetic average approach.
BidderCriteriaTotal scoreActual winnerC1C2C3C4C5Bidder 118.020.520.217.57.884.0YesBidder 213.022.021.017.08.081.0Bidder 318.319.718.816.87.881.5Bidder 412.518.517.317.27.272.7Bidder 512.721.320.116.17.778.0Bidder 617.420.119.117.08.081.7YesBidder 716.919.918.917.08.080.6


#### 4.2.2. Solution by the DEA Method

According to the data in Table9, the DEAP software was used to run the DEA CCR model. The security services company selection evaluation results are listed in Table 11. In Table 11, a bidder’s DEA value of 100% indicates that the bidder’s performance is the highest.Table 11
Security services company DEA evaluation results.
DMUOutput 1Output 2Output 3Output 4Output 5InputDEA value (%)Actual winnerDMU 118.020.520.217.57.81.0100.0YesDMU 213.022.021.017.08.01.0100.0YesDMU 318.319.718.816.87.81.0100.0YesDMU 412.518.517.317.27.21.098.3DMU 512.721.320.116.17.71.096.9DMU 617.420.119.117.08.01.0100.0YesDMU 716.919.918.917.08.01.0100.0Yes


#### 4.2.3. Solution by the Proposed Method

According to the results of Table9, equations (1) and (2) were used to calculate the normalized decision matrix of different security companies, as listed in Table 12.Table 12
Normalized the decision matrix of different security companies.
BidderCriteriaC1C2C3C4C5Bidder 10.9480.5710.7841.0000.750Bidder 20.0861.0001.0000.6431.000Bidder 31.0000.3430.4050.5000.750Bidder 40.0000.0000.0000.7860.000Bidder 50.0340.8000.7570.0000.625Bidder 60.8450.4570.4860.6431.000Bidder 70.7590.4000.4320.6431.000According to the results of Table12, equations (3)–(5) were used to calculate the preference function Pja,b and aggregated preference function for different security companies, as shown in Table 13.Table 13
Aggregated preference function of different security companies.
BidderBidder 1Bidder 2Bidder 3Bidder 4Bidder 5Bidder 6Bidder 7Bidder 1—0.2440.2520.6460.4020.1950.240Bidder 20.186—0.3670.6170.2870.2640.292Bidder 30.0100.183—0.4620.3060.0310.048Bidder 40.0000.0290.057—0.1570.0290.029Bidder 50.0570.0000.2020.459—0.1530.181Bidder 60.0250.1520.1020.5050.328—0.045Bidder 70.0250.1340.0750.4600.3110.000—According to the results of Table13, equations (6)–(8) were used to calculate the entering flow, leaving flow, and net outranking flow for different security companies, as listed in Table 14.Table 14
Entering, leaving, and net flows for different security companies.
BidderEntering flowLeaving flowNet outranking flowBidder 10.0510.3300.279Bidder 20.1240.3360.212Bidder 30.1760.173–0.002Bidder 40.5250.050–0.475Bidder 50.2990.175–0.123Bidder 60.1120.1930.081Bidder 70.1390.1670.028


#### 4.2.4. Summary

Case Project 2 involved the selection of two security companies to provide security for a science park in Taiwan. Table15 displays the calculation results of the traditional weighted arithmetic averaging method, the DEA method, and the proposed method for Case Project 2.Table 15
Results of different calculation methods for Case Project 2.
BidderTraditional weighted arithmetic averaging methodDEA methodThe proposed methodTotal scoreRankingDEA value (%)RankingNet outranking flowRankingBidder 184.01100.010.2791Bidder 281.04100.010.2122Bidder 381.53100.01–0.0025Bidder 472.7798.36–0.4757Bidder 578.0696.97–0.1236Bidder 681.72100.010.0813Bidder 780.65100.010.0284In Table15, the traditional weighted arithmetic averaging method identifies bidders 1 and 6 as having the two highest scores, thus the winning bidders. However, the traditional weighted arithmetic averaging method does not consider the relative importance of the criteria or subjective preferences of the experts, which will result in an incorrect conclusion. From Table 15, the DEA values for bidders 1, 2, 3, 6, and 7 are 100%. The DEA method thus has a high duplicate rate problem. The proposed extended PROMETHEE II method simultaneously considers the objective weights of the assessment criteria and the subjective preferences of the experts. Thus, the proposed method identifies the top two net outranking flows as 0.279 (bidder 1) and 0.212 (bidder 2), which are the winning bids. Therefore, the proposed extended PROMETHEE II approach is more suitable for solving the contractor selection problem.


### 4.3. Case Project 3

Case Project 3 involved the selection of the best contractor for a local Koji pottery industry promotion company in Chiayi, Taiwan [42]. In this local Koji pottery industry promotion case, there were three bidders, and the final evaluation committee consisted of four committee members. The assessment criteria included the content of service proposal (C1), service management and control of the project (C2), bidder’s bid price and components (C3), project ideas and feedback (C4), and brief and responsive (C5). The relative weights for the five assessment criteria were 0.30, 0.25, 0.15, 0.20, and 0.10, respectively. Performance evaluation scores of each bidder are listed in Table 16.Table 16
Performance evaluation scores of local Koji pottery industry promotion contractors.
CriterionWeighting (%)ExpertBidder 1Bidder 2Bidder 3C130P1252425P2242525P3242324P4242524C225P1232322P2222221P3∗∗∗P4212021C315P1111312P2111212P3121312P4121312C420P1171817P2161617P3151618P4151717C510P1878P2667P3778P4877∗indicates missing or nonexistent data.


#### 4.3.1. Solution by the Traditional Weighted Arithmetic Average Approach

Some information provided by Expert P3 were missing or nonexistent data, and thus, only information provided by Experts P1, P2, and P4 were considered. According to the results of Table16, equation (9) was used to calculate the bidder scores for local Koji pottery industry promotion contractor selection, as shown in Table 17.Table 17
Local Koji pottery industry promotion contractor selection by the traditional weighted arithmetic averaging method.
BidderExpertCriteriaTotal scoreActual winnerC1C2C3C4C5Bidder 1P125231117881.0P2242211166P4242112158Bidder 2P124231318782.7YesP2252212166P4252013177Bidder 3P125221217882.3P2252112177P4242112177


#### 4.3.2. Solution by the DEA Method

From Table16, Experts P1, P2, and P4 provided complete evaluation information, while Expert P3’s evaluation information was incomplete. The DEA method can only work with complete expert information. Therefore, only the information provided by Experts P1, P2, and P4 were considered. According to Table 16, the DEAP software was used to run the DEA CCR model, and the local Koji pottery industry promotion company selection evaluation result is shown in Table 18.Table 18
DEA local Koji pottery industry promotion contractor evaluation result.
BidderOutput 1Output 2Output 3Output 4Output 5InputDEA value (%)Actual winnerBidder 124.322.011.316.07.31.0100.0YesBidder 224.721.712.717.06.71.0100.0YesBidder 324.721.312.017.07.31.0100.0Yes


#### 4.3.3. Solution by the Proposed Method

The proposed novel contractor selection technique is not only able to handle the objective weights of assessment criteria and subjective preferences of experts but can also handle missing or nonexistent contractor selection assessment process data by using the average value of the complete information to complete the information. According to Table16, equations (1) and (2) were used to compute normalized decision matrix for the local Koji pottery industry promotion companies, as listed in Table 19.Table 19
Normalized the decision matrix for different promotion contractors.
BidderCriteriaC1C2C3C4C5Bidder 10.0001.0000.0000.0000.667Bidder 20.0000.5001.0000.6670.000Bidder 31.0000.0000.4001.0001.000According to Table19, quations (3)–(5) were used to calculate the preference function Pja,b and aggregated preference function for different local Koji pottery industry promotion companies, as shown in Table 20.Table 20
Aggregated preference function for different promotion contractors.
BidderBidder 1Bidder 2Bidder 3Bidder 1—0.1920.250Bidder 20.283—0.215Bidder 30.5930.467—According to Table20, equations (6)–(8) were used to calculate the entering flow, leaving flow, and net outranking flow of different local Koji pottery industry promotion companies, as shown in Table 21.Table 21
Entering, leaving, and net flows of different promotion contractors.
BidderEntering flowLeaving flowNet outranking flowBidder 10.4380.221−0.218Bidder 20.3290.249−0.080Bidder 30.2330.5300.298


#### 4.3.4. Summary

Case Project 3 involved the selection of the best bidder as a local Koji pottery industry promotion company in Taiwan. Table22 shows the calculation results of the traditional weighted arithmetic averaging method, the DEA method, and the proposed method for Case Project 3. From Table 22, the DEA values of bidders 1, 2, and 3 are 100%, which illustrates the high repetition rate problem of the DEA method. Moreover, neither the traditional weighted arithmetic averaging method nor the DEA method can handle incomplete contractor selection process information. Therefore, the proposed method is more a general contractor selection technique and better suited for handling real-world problems.Table 22
Results of different calculation methods for Case Project 3.
BidderTraditional weighted arithmetic averaging methodDEA methodThe proposed methodTotal scoreRankingDEA value (%)RankingNet outranking flowRankingBidder 181.03100.01−0.2183Bidder 282.71100.01−0.0802Bidder 382.32100.010.2981


## 4.1. Case Project 1

This section presents an illustrative example of software supplier selection adapted from Yang et al. [2] who proved the effectiveness of the proposed novel contractor selection approach. The case company must select the optimal software supplier in order to develop and plan the management information system. There were six bidders, and a bid opening committee that consisted of four committee members in this software supplier case. The assessment criteria included the comprehension degree for the project (C1), system design and planning (C2), coding ability of a system (C3), management ability of project (C4), and past achievements and experience (C5). The relative weights of the five evaluation criteria were 0.10, 0.30, 0.30, 0.20, and 0.10, respectively. The five evaluation criteria were then used by the final bid opening committee to determine the most suitable software supplier. The five assessment criteria scores for each bidder are listed in Table 1.Table 1
Actual evaluation data of software supplier.
BidderCriteriaC1C2C3C4C5Bidder 16.223.823.115.67.2Bidder 26.023.023.015.06.3Bidder 38.026.326.317.58.5Bidder 46.721.522.314.86.0Bidder 58.826.027.017.08.5Bidder 68.224.526.316.78.0


### 4.1.1. Solution by the Traditional Weighted Arithmetic Averaging Approach

The traditional weighted arithmetic averaging approach is one of the simplest and most commonly used aggregation operators, and it can be explained as follows.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 and ∑i=1nwi=1,wi∈0,1 as follows:(9)WWAa1,a2,…,an=∑i=1nwiai,where ai is the argument variable.
According to the results of Table1, equation (9) was used to calculate the bidder scores for software supplier selection, as shown in Table 2.Table 2
Software supplier selection by the traditional weighted arithmetic averaging method.
BidderCriteriaTotal scoreActual winnerC1C2C3C4C5Bidder 16.223.823.115.67.275.9Bidder 26.023.023.015.06.373.3Bidder 38.026.326.317.58.587.6YesBidder 46.721.522.314.86.071.3Bidder 58.826.027.017.08.587.0Bidder 68.224.526.316.78.084.7


### 4.1.2. Solution by the DEA Method

Yang et al. [2] proposed a DEA model with single input and multiple outputs to solve the best-value contractor selection problem. They used the decision-making unit (DMU) as a bidder, and the DEA input values were equal to 1. According to Table 1, the DEAP software was used to run the DEA CCR model, and the software supplier evaluation results as listed in Table 3, where a bidder’s DEA value of 100% indicates that the bidder has the highest comparative efficiency.Table 3
Software supplier DEA evaluation results.
DMUOutput 1Output 2Output 3Output 4Output 5InputDEA value (%)Actual winnerDMU 16.223.823.115.67.21.090.5DMU 26.023.023.015.06.31.087.5DMU 38.026.326.317.58.51.0100.0YesDMU 46.721.522.314.86.01.084.7DMU 68.826.027.017.08.51.0100.0YesDMU 68.224.526.316.78.01.097.8


### 4.1.3. Solution by the Proposed Novel Contractor Selection Approach

The proposed novel contractor selection approach integrates the PROMETHEE II method and the soft set approach to select suitable contractors. In the software supplier selection example, the five assessment criteria scores for six bidders in the final bid opening committee are as listed in Table1 (Steps 1 ∼ 4). The following procedure describes the remaining steps of the proposed method:Step 5: normalize the decision matrix.According to Table1, equations (1) and (2) were used to calculate the normalized decision matrix for different software suppliers, as listed in Table 4.Step 6: compute the preference functionPja,b.Step 7: compute the aggregated preference functionπa,b.According to Table4, equations (3)–(5) were used to compute the preference function (Pja,b) and aggregated preference function (πa,b) for different software suppliers, as listed in Table 5.Step 8: compute the entering flow and leaving flow for each alternative.According to Table5, equations (6) and (7) were used to calculate the entering and leaving flows for different software suppliers, as shown in Table 6.Step 9: determine the ranking of all possible alternatives.Table 4
Normalized the decision matrix for different software suppliers.
BidderCriteriaC1C2C3C4C5Bidder 10.0710.4790.1700.2960.480Bidder 20.0000.3130.1490.0740.120Bidder 30.7141.0000.8511.0001.000Bidder 40.2500.0000.0000.0000.000Bidder 51.0000.9381.0000.8151.000Bidder 60.7860.6250.8510.7040.800Table 5
Aggregated preference function for different software suppliers.
BidderBidder 1Bidder 2Bidder 3Bidder 4Bidder 5Bidder 6Bidder 1—0.1440.0000.3020.0000.000Bidder 20.000—0.0000.1650.0000.000Bidder 30.6180.762—0.9020.0560.192Bidder 40.0180.0250.000—0.0000.000Bidder 50.6350.7790.0730.919—0.202Bidder 60.4330.5770.0070.7170.000—Table 6
Entering and leaving flows for different software suppliers.
BidderEntering flowLeaving flowBidder 10.3410.089Bidder 20.4570.033Bidder 30.0160.506Bidder 40.6010.009Bidder 50.0110.522Bidder 60.0790.347According to Table6, equation (8) was used to calculate the net flow for different software suppliers. The net flow values φa was ranked from highest to lowest, with a higher net flow value φa representing a better software supplier, as listed in Table 7.Table 7
Net outranking flow values for different alternative software suppliers.
BidderNet outranking flowRankBidder 1−0.2514Bidder 2−0.4245Bidder 30.4902Bidder 4−0.5936Bidder 50.5111Bidder 60.2683


### 4.1.4. Summary

Case Project 1 involves selecting a software supplier to develop a management information system for the case company. The input data for each bidder was listed in Table1. Table 8 lists the different calculation results of the traditional weighted arithmetic averaging method, the DEA method, and the proposed method for Case Project 1.Table 8
Results of different calculation methods for Case Project 1.
BidderTraditional weighted arithmetic averaging methodDEA methodThe proposed methodTotal scoreRankingDEA value (%)RankingNet outranking flowRankingBidder 175.9490.54–0.2514Bidder 273.3587.55–0.4245Bidder 387.61100.010.4902Bidder 471.3684.76–0.5936Bidder 587.02100.010.5111Bidder 684.7397.830.2683From Table8, the DEA values for bidders 3 and 5 were 100%. The proposed extended PROMETHEE II method can solve this problem of duplicate DEA value. In the proposed method, the net outranking flow of bidders 3 and 5 were 0.490 and 0.511, respectively. Therefore, the proposed method can better sort bidder scores.


## 4.1.1. Solution by the Traditional Weighted Arithmetic Averaging Approach

The traditional weighted arithmetic averaging approach is one of the simplest and most commonly used aggregation operators, and it can be explained as follows.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 and ∑i=1nwi=1,wi∈0,1 as follows:(9)WWAa1,a2,…,an=∑i=1nwiai,where ai is the argument variable.
According to the results of Table1, equation (9) was used to calculate the bidder scores for software supplier selection, as shown in Table 2.Table 2
Software supplier selection by the traditional weighted arithmetic averaging method.
BidderCriteriaTotal scoreActual winnerC1C2C3C4C5Bidder 16.223.823.115.67.275.9Bidder 26.023.023.015.06.373.3Bidder 38.026.326.317.58.587.6YesBidder 46.721.522.314.86.071.3Bidder 58.826.027.017.08.587.0Bidder 68.224.526.316.78.084.7


## 4.1.2. Solution by the DEA Method

Yang et al. [2] proposed a DEA model with single input and multiple outputs to solve the best-value contractor selection problem. They used the decision-making unit (DMU) as a bidder, and the DEA input values were equal to 1. According to Table 1, the DEAP software was used to run the DEA CCR model, and the software supplier evaluation results as listed in Table 3, where a bidder’s DEA value of 100% indicates that the bidder has the highest comparative efficiency.Table 3
Software supplier DEA evaluation results.
DMUOutput 1Output 2Output 3Output 4Output 5InputDEA value (%)Actual winnerDMU 16.223.823.115.67.21.090.5DMU 26.023.023.015.06.31.087.5DMU 38.026.326.317.58.51.0100.0YesDMU 46.721.522.314.86.01.084.7DMU 68.826.027.017.08.51.0100.0YesDMU 68.224.526.316.78.01.097.8


## 4.1.3. Solution by the Proposed Novel Contractor Selection Approach

The proposed novel contractor selection approach integrates the PROMETHEE II method and the soft set approach to select suitable contractors. In the software supplier selection example, the five assessment criteria scores for six bidders in the final bid opening committee are as listed in Table1 (Steps 1 ∼ 4). The following procedure describes the remaining steps of the proposed method:Step 5: normalize the decision matrix.According to Table1, equations (1) and (2) were used to calculate the normalized decision matrix for different software suppliers, as listed in Table 4.Step 6: compute the preference functionPja,b.Step 7: compute the aggregated preference functionπa,b.According to Table4, equations (3)–(5) were used to compute the preference function (Pja,b) and aggregated preference function (πa,b) for different software suppliers, as listed in Table 5.Step 8: compute the entering flow and leaving flow for each alternative.According to Table5, equations (6) and (7) were used to calculate the entering and leaving flows for different software suppliers, as shown in Table 6.Step 9: determine the ranking of all possible alternatives.Table 4
Normalized the decision matrix for different software suppliers.
BidderCriteriaC1C2C3C4C5Bidder 10.0710.4790.1700.2960.480Bidder 20.0000.3130.1490.0740.120Bidder 30.7141.0000.8511.0001.000Bidder 40.2500.0000.0000.0000.000Bidder 51.0000.9381.0000.8151.000Bidder 60.7860.6250.8510.7040.800Table 5
Aggregated preference function for different software suppliers.
BidderBidder 1Bidder 2Bidder 3Bidder 4Bidder 5Bidder 6Bidder 1—0.1440.0000.3020.0000.000Bidder 20.000—0.0000.1650.0000.000Bidder 30.6180.762—0.9020.0560.192Bidder 40.0180.0250.000—0.0000.000Bidder 50.6350.7790.0730.919—0.202Bidder 60.4330.5770.0070.7170.000—Table 6
Entering and leaving flows for different software suppliers.
BidderEntering flowLeaving flowBidder 10.3410.089Bidder 20.4570.033Bidder 30.0160.506Bidder 40.6010.009Bidder 50.0110.522Bidder 60.0790.347According to Table6, equation (8) was used to calculate the net flow for different software suppliers. The net flow values φa was ranked from highest to lowest, with a higher net flow value φa representing a better software supplier, as listed in Table 7.Table 7
Net outranking flow values for different alternative software suppliers.
BidderNet outranking flowRankBidder 1−0.2514Bidder 2−0.4245Bidder 30.4902Bidder 4−0.5936Bidder 50.5111Bidder 60.2683


## 4.1.4. Summary

Case Project 1 involves selecting a software supplier to develop a management information system for the case company. The input data for each bidder was listed in Table1. Table 8 lists the different calculation results of the traditional weighted arithmetic averaging method, the DEA method, and the proposed method for Case Project 1.Table 8
Results of different calculation methods for Case Project 1.
BidderTraditional weighted arithmetic averaging methodDEA methodThe proposed methodTotal scoreRankingDEA value (%)RankingNet outranking flowRankingBidder 175.9490.54–0.2514Bidder 273.3587.55–0.4245Bidder 387.61100.010.4902Bidder 471.3684.76–0.5936Bidder 587.02100.010.5111Bidder 684.7397.830.2683From Table8, the DEA values for bidders 3 and 5 were 100%. The proposed extended PROMETHEE II method can solve this problem of duplicate DEA value. In the proposed method, the net outranking flow of bidders 3 and 5 were 0.490 and 0.511, respectively. Therefore, the proposed method can better sort bidder scores.


## 4.2. Case Project 2

Case Project 2 involves the selection of two security companies to provide security services for a science park [2]. There were seven bidders competing for two awards in this case. The assessment criteria included company organization (C1), planning feasibility (C2), professional capability (C3), price (C4), and presentation and question response (C5). The relative weights for the five assessment criteria were 0.20, 0.25, 0.25, 0.20, and 0.10, respectively, and the final evaluation committee selected the best two security services companies based on the five assessment criteria. The five assessment criteria scores for each security services company are listed in Table 9.Table 9
Actual evaluation data of security services companies.
BidderCriteriaC1C2C3C4C5Bidder 118.020.520.217.57.8Bidder 213.022.021.017.08.0Bidder 318.319.718.816.87.8Bidder 412.518.517.317.27.2Bidder 512.721.320.116.17.7Bidder 617.420.119.117.08.0Bidder 716.919.918.917.08.0


### 4.2.1. Solution by the Traditional Weighted Arithmetic Average Approach

Based on the results of Table9, equation (9) was used to calculate the bidder scores for security services company selection, as shown in Table 10.Table 10
Security services company selection by the traditional weighted arithmetic average approach.
BidderCriteriaTotal scoreActual winnerC1C2C3C4C5Bidder 118.020.520.217.57.884.0YesBidder 213.022.021.017.08.081.0Bidder 318.319.718.816.87.881.5Bidder 412.518.517.317.27.272.7Bidder 512.721.320.116.17.778.0Bidder 617.420.119.117.08.081.7YesBidder 716.919.918.917.08.080.6


### 4.2.2. Solution by the DEA Method

According to the data in Table9, the DEAP software was used to run the DEA CCR model. The security services company selection evaluation results are listed in Table 11. In Table 11, a bidder’s DEA value of 100% indicates that the bidder’s performance is the highest.Table 11
Security services company DEA evaluation results.
DMUOutput 1Output 2Output 3Output 4Output 5InputDEA value (%)Actual winnerDMU 118.020.520.217.57.81.0100.0YesDMU 213.022.021.017.08.01.0100.0YesDMU 318.319.718.816.87.81.0100.0YesDMU 412.518.517.317.27.21.098.3DMU 512.721.320.116.17.71.096.9DMU 617.420.119.117.08.01.0100.0YesDMU 716.919.918.917.08.01.0100.0Yes


### 4.2.3. Solution by the Proposed Method

According to the results of Table9, equations (1) and (2) were used to calculate the normalized decision matrix of different security companies, as listed in Table 12.Table 12
Normalized the decision matrix of different security companies.
BidderCriteriaC1C2C3C4C5Bidder 10.9480.5710.7841.0000.750Bidder 20.0861.0001.0000.6431.000Bidder 31.0000.3430.4050.5000.750Bidder 40.0000.0000.0000.7860.000Bidder 50.0340.8000.7570.0000.625Bidder 60.8450.4570.4860.6431.000Bidder 70.7590.4000.4320.6431.000According to the results of Table12, equations (3)–(5) were used to calculate the preference function Pja,b and aggregated preference function for different security companies, as shown in Table 13.Table 13
Aggregated preference function of different security companies.
BidderBidder 1Bidder 2Bidder 3Bidder 4Bidder 5Bidder 6Bidder 7Bidder 1—0.2440.2520.6460.4020.1950.240Bidder 20.186—0.3670.6170.2870.2640.292Bidder 30.0100.183—0.4620.3060.0310.048Bidder 40.0000.0290.057—0.1570.0290.029Bidder 50.0570.0000.2020.459—0.1530.181Bidder 60.0250.1520.1020.5050.328—0.045Bidder 70.0250.1340.0750.4600.3110.000—According to the results of Table13, equations (6)–(8) were used to calculate the entering flow, leaving flow, and net outranking flow for different security companies, as listed in Table 14.Table 14
Entering, leaving, and net flows for different security companies.
BidderEntering flowLeaving flowNet outranking flowBidder 10.0510.3300.279Bidder 20.1240.3360.212Bidder 30.1760.173–0.002Bidder 40.5250.050–0.475Bidder 50.2990.175–0.123Bidder 60.1120.1930.081Bidder 70.1390.1670.028


### 4.2.4. Summary

Case Project 2 involved the selection of two security companies to provide security for a science park in Taiwan. Table15 displays the calculation results of the traditional weighted arithmetic averaging method, the DEA method, and the proposed method for Case Project 2.Table 15
Results of different calculation methods for Case Project 2.
BidderTraditional weighted arithmetic averaging methodDEA methodThe proposed methodTotal scoreRankingDEA value (%)RankingNet outranking flowRankingBidder 184.01100.010.2791Bidder 281.04100.010.2122Bidder 381.53100.01–0.0025Bidder 472.7798.36–0.4757Bidder 578.0696.97–0.1236Bidder 681.72100.010.0813Bidder 780.65100.010.0284In Table15, the traditional weighted arithmetic averaging method identifies bidders 1 and 6 as having the two highest scores, thus the winning bidders. However, the traditional weighted arithmetic averaging method does not consider the relative importance of the criteria or subjective preferences of the experts, which will result in an incorrect conclusion. From Table 15, the DEA values for bidders 1, 2, 3, 6, and 7 are 100%. The DEA method thus has a high duplicate rate problem. The proposed extended PROMETHEE II method simultaneously considers the objective weights of the assessment criteria and the subjective preferences of the experts. Thus, the proposed method identifies the top two net outranking flows as 0.279 (bidder 1) and 0.212 (bidder 2), which are the winning bids. Therefore, the proposed extended PROMETHEE II approach is more suitable for solving the contractor selection problem.


## 4.2.1. Solution by the Traditional Weighted Arithmetic Average Approach

Based on the results of Table9, equation (9) was used to calculate the bidder scores for security services company selection, as shown in Table 10.Table 10
Security services company selection by the traditional weighted arithmetic average approach.
BidderCriteriaTotal scoreActual winnerC1C2C3C4C5Bidder 118.020.520.217.57.884.0YesBidder 213.022.021.017.08.081.0Bidder 318.319.718.816.87.881.5Bidder 412.518.517.317.27.272.7Bidder 512.721.320.116.17.778.0Bidder 617.420.119.117.08.081.7YesBidder 716.919.918.917.08.080.6


## 4.2.2. Solution by the DEA Method

According to the data in Table9, the DEAP software was used to run the DEA CCR model. The security services company selection evaluation results are listed in Table 11. In Table 11, a bidder’s DEA value of 100% indicates that the bidder’s performance is the highest.Table 11
Security services company DEA evaluation results.
DMUOutput 1Output 2Output 3Output 4Output 5InputDEA value (%)Actual winnerDMU 118.020.520.217.57.81.0100.0YesDMU 213.022.021.017.08.01.0100.0YesDMU 318.319.718.816.87.81.0100.0YesDMU 412.518.517.317.27.21.098.3DMU 512.721.320.116.17.71.096.9DMU 617.420.119.117.08.01.0100.0YesDMU 716.919.918.917.08.01.0100.0Yes


## 4.2.3. Solution by the Proposed Method

According to the results of Table9, equations (1) and (2) were used to calculate the normalized decision matrix of different security companies, as listed in Table 12.Table 12
Normalized the decision matrix of different security companies.
BidderCriteriaC1C2C3C4C5Bidder 10.9480.5710.7841.0000.750Bidder 20.0861.0001.0000.6431.000Bidder 31.0000.3430.4050.5000.750Bidder 40.0000.0000.0000.7860.000Bidder 50.0340.8000.7570.0000.625Bidder 60.8450.4570.4860.6431.000Bidder 70.7590.4000.4320.6431.000According to the results of Table12, equations (3)–(5) were used to calculate the preference function Pja,b and aggregated preference function for different security companies, as shown in Table 13.Table 13
Aggregated preference function of different security companies.
BidderBidder 1Bidder 2Bidder 3Bidder 4Bidder 5Bidder 6Bidder 7Bidder 1—0.2440.2520.6460.4020.1950.240Bidder 20.186—0.3670.6170.2870.2640.292Bidder 30.0100.183—0.4620.3060.0310.048Bidder 40.0000.0290.057—0.1570.0290.029Bidder 50.0570.0000.2020.459—0.1530.181Bidder 60.0250.1520.1020.5050.328—0.045Bidder 70.0250.1340.0750.4600.3110.000—According to the results of Table13, equations (6)–(8) were used to calculate the entering flow, leaving flow, and net outranking flow for different security companies, as listed in Table 14.Table 14
Entering, leaving, and net flows for different security companies.
BidderEntering flowLeaving flowNet outranking flowBidder 10.0510.3300.279Bidder 20.1240.3360.212Bidder 30.1760.173–0.002Bidder 40.5250.050–0.475Bidder 50.2990.175–0.123Bidder 60.1120.1930.081Bidder 70.1390.1670.028


## 4.2.4. Summary

Case Project 2 involved the selection of two security companies to provide security for a science park in Taiwan. Table15 displays the calculation results of the traditional weighted arithmetic averaging method, the DEA method, and the proposed method for Case Project 2.Table 15
Results of different calculation methods for Case Project 2.
BidderTraditional weighted arithmetic averaging methodDEA methodThe proposed methodTotal scoreRankingDEA value (%)RankingNet outranking flowRankingBidder 184.01100.010.2791Bidder 281.04100.010.2122Bidder 381.53100.01–0.0025Bidder 472.7798.36–0.4757Bidder 578.0696.97–0.1236Bidder 681.72100.010.0813Bidder 780.65100.010.0284In Table15, the traditional weighted arithmetic averaging method identifies bidders 1 and 6 as having the two highest scores, thus the winning bidders. However, the traditional weighted arithmetic averaging method does not consider the relative importance of the criteria or subjective preferences of the experts, which will result in an incorrect conclusion. From Table 15, the DEA values for bidders 1, 2, 3, 6, and 7 are 100%. The DEA method thus has a high duplicate rate problem. The proposed extended PROMETHEE II method simultaneously considers the objective weights of the assessment criteria and the subjective preferences of the experts. Thus, the proposed method identifies the top two net outranking flows as 0.279 (bidder 1) and 0.212 (bidder 2), which are the winning bids. Therefore, the proposed extended PROMETHEE II approach is more suitable for solving the contractor selection problem.


## 4.3. Case Project 3

Case Project 3 involved the selection of the best contractor for a local Koji pottery industry promotion company in Chiayi, Taiwan [42]. In this local Koji pottery industry promotion case, there were three bidders, and the final evaluation committee consisted of four committee members. The assessment criteria included the content of service proposal (C1), service management and control of the project (C2), bidder’s bid price and components (C3), project ideas and feedback (C4), and brief and responsive (C5). The relative weights for the five assessment criteria were 0.30, 0.25, 0.15, 0.20, and 0.10, respectively. Performance evaluation scores of each bidder are listed in Table 16.Table 16
Performance evaluation scores of local Koji pottery industry promotion contractors.
CriterionWeighting (%)ExpertBidder 1Bidder 2Bidder 3C130P1252425P2242525P3242324P4242524C225P1232322P2222221P3∗∗∗P4212021C315P1111312P2111212P3121312P4121312C420P1171817P2161617P3151618P4151717C510P1878P2667P3778P4877∗indicates missing or nonexistent data.


### 4.3.1. Solution by the Traditional Weighted Arithmetic Average Approach

Some information provided by Expert P3 were missing or nonexistent data, and thus, only information provided by Experts P1, P2, and P4 were considered. According to the results of Table16, equation (9) was used to calculate the bidder scores for local Koji pottery industry promotion contractor selection, as shown in Table 17.Table 17
Local Koji pottery industry promotion contractor selection by the traditional weighted arithmetic averaging method.
BidderExpertCriteriaTotal scoreActual winnerC1C2C3C4C5Bidder 1P125231117881.0P2242211166P4242112158Bidder 2P124231318782.7YesP2252212166P4252013177Bidder 3P125221217882.3P2252112177P4242112177


### 4.3.2. Solution by the DEA Method

From Table16, Experts P1, P2, and P4 provided complete evaluation information, while Expert P3’s evaluation information was incomplete. The DEA method can only work with complete expert information. Therefore, only the information provided by Experts P1, P2, and P4 were considered. According to Table 16, the DEAP software was used to run the DEA CCR model, and the local Koji pottery industry promotion company selection evaluation result is shown in Table 18.Table 18
DEA local Koji pottery industry promotion contractor evaluation result.
BidderOutput 1Output 2Output 3Output 4Output 5InputDEA value (%)Actual winnerBidder 124.322.011.316.07.31.0100.0YesBidder 224.721.712.717.06.71.0100.0YesBidder 324.721.312.017.07.31.0100.0Yes


### 4.3.3. Solution by the Proposed Method

The proposed novel contractor selection technique is not only able to handle the objective weights of assessment criteria and subjective preferences of experts but can also handle missing or nonexistent contractor selection assessment process data by using the average value of the complete information to complete the information. According to Table16, equations (1) and (2) were used to compute normalized decision matrix for the local Koji pottery industry promotion companies, as listed in Table 19.Table 19
Normalized the decision matrix for different promotion contractors.
BidderCriteriaC1C2C3C4C5Bidder 10.0001.0000.0000.0000.667Bidder 20.0000.5001.0000.6670.000Bidder 31.0000.0000.4001.0001.000According to Table19, quations (3)–(5) were used to calculate the preference function Pja,b and aggregated preference function for different local Koji pottery industry promotion companies, as shown in Table 20.Table 20
Aggregated preference function for different promotion contractors.
BidderBidder 1Bidder 2Bidder 3Bidder 1—0.1920.250Bidder 20.283—0.215Bidder 30.5930.467—According to Table20, equations (6)–(8) were used to calculate the entering flow, leaving flow, and net outranking flow of different local Koji pottery industry promotion companies, as shown in Table 21.Table 21
Entering, leaving, and net flows of different promotion contractors.
BidderEntering flowLeaving flowNet outranking flowBidder 10.4380.221−0.218Bidder 20.3290.249−0.080Bidder 30.2330.5300.298


### 4.3.4. Summary

Case Project 3 involved the selection of the best bidder as a local Koji pottery industry promotion company in Taiwan. Table22 shows the calculation results of the traditional weighted arithmetic averaging method, the DEA method, and the proposed method for Case Project 3. From Table 22, the DEA values of bidders 1, 2, and 3 are 100%, which illustrates the high repetition rate problem of the DEA method. Moreover, neither the traditional weighted arithmetic averaging method nor the DEA method can handle incomplete contractor selection process information. Therefore, the proposed method is more a general contractor selection technique and better suited for handling real-world problems.Table 22
Results of different calculation methods for Case Project 3.
BidderTraditional weighted arithmetic averaging methodDEA methodThe proposed methodTotal scoreRankingDEA value (%)RankingNet outranking flowRankingBidder 181.03100.01−0.2183Bidder 282.71100.01−0.0802Bidder 382.32100.010.2981


## 4.3.1. Solution by the Traditional Weighted Arithmetic Average Approach

Some information provided by Expert P3 were missing or nonexistent data, and thus, only information provided by Experts P1, P2, and P4 were considered. According to the results of Table16, equation (9) was used to calculate the bidder scores for local Koji pottery industry promotion contractor selection, as shown in Table 17.Table 17
Local Koji pottery industry promotion contractor selection by the traditional weighted arithmetic averaging method.
BidderExpertCriteriaTotal scoreActual winnerC1C2C3C4C5Bidder 1P125231117881.0P2242211166P4242112158Bidder 2P124231318782.7YesP2252212166P4252013177Bidder 3P125221217882.3P2252112177P4242112177


## 4.3.2. Solution by the DEA Method

From Table16, Experts P1, P2, and P4 provided complete evaluation information, while Expert P3’s evaluation information was incomplete. The DEA method can only work with complete expert information. Therefore, only the information provided by Experts P1, P2, and P4 were considered. According to Table 16, the DEAP software was used to run the DEA CCR model, and the local Koji pottery industry promotion company selection evaluation result is shown in Table 18.Table 18
DEA local Koji pottery industry promotion contractor evaluation result.
BidderOutput 1Output 2Output 3Output 4Output 5InputDEA value (%)Actual winnerBidder 124.322.011.316.07.31.0100.0YesBidder 224.721.712.717.06.71.0100.0YesBidder 324.721.312.017.07.31.0100.0Yes


## 4.3.3. Solution by the Proposed Method

The proposed novel contractor selection technique is not only able to handle the objective weights of assessment criteria and subjective preferences of experts but can also handle missing or nonexistent contractor selection assessment process data by using the average value of the complete information to complete the information. According to Table16, equations (1) and (2) were used to compute normalized decision matrix for the local Koji pottery industry promotion companies, as listed in Table 19.Table 19
Normalized the decision matrix for different promotion contractors.
BidderCriteriaC1C2C3C4C5Bidder 10.0001.0000.0000.0000.667Bidder 20.0000.5001.0000.6670.000Bidder 31.0000.0000.4001.0001.000According to Table19, quations (3)–(5) were used to calculate the preference function Pja,b and aggregated preference function for different local Koji pottery industry promotion companies, as shown in Table 20.Table 20
Aggregated preference function for different promotion contractors.
BidderBidder 1Bidder 2Bidder 3Bidder 1—0.1920.250Bidder 20.283—0.215Bidder 30.5930.467—According to Table20, equations (6)–(8) were used to calculate the entering flow, leaving flow, and net outranking flow of different local Koji pottery industry promotion companies, as shown in Table 21.Table 21
Entering, leaving, and net flows of different promotion contractors.
BidderEntering flowLeaving flowNet outranking flowBidder 10.4380.221−0.218Bidder 20.3290.249−0.080Bidder 30.2330.5300.298


## 4.3.4. Summary

Case Project 3 involved the selection of the best bidder as a local Koji pottery industry promotion company in Taiwan. Table22 shows the calculation results of the traditional weighted arithmetic averaging method, the DEA method, and the proposed method for Case Project 3. From Table 22, the DEA values of bidders 1, 2, and 3 are 100%, which illustrates the high repetition rate problem of the DEA method. Moreover, neither the traditional weighted arithmetic averaging method nor the DEA method can handle incomplete contractor selection process information. Therefore, the proposed method is more a general contractor selection technique and better suited for handling real-world problems.Table 22
Results of different calculation methods for Case Project 3.
BidderTraditional weighted arithmetic averaging methodDEA methodThe proposed methodTotal scoreRankingDEA value (%)RankingNet outranking flowRankingBidder 181.03100.01−0.2183Bidder 282.71100.01−0.0802Bidder 382.32100.010.2981


## 5. Conclusions

Contractor selection is a critical component of any company’s pursuit of sustainable development, in addition, choosing suitable contractors ensures that project management plans are executed as expected. However, contractor selection includes qualitative and quantitative assessment criteria at the same time, which becomes a complicated MCDM problem. Traditional contractor selection methods are unable to handle nonexistent or missing assessment criteria data nor consider the relative importance of criteria in the contractor selection assessment process. In order to effectively address this problem, this study extended the PROMETHEE II method to propose a novel contractor selection technique. Moreover, three numerical examples are applied to prove the correctness and effectiveness of the proposed technique. The simulation results showed that the proposed extended PROMETHEE II approach is a more general contractor selection technique for handling incomplete information than the traditional weighted arithmetic averaging method and the DEA method.The main advantages of the proposed extended PROMETHEE II approach are as follows:(1)
The proposed approach can handle incomplete assessment criteria information(2)
The proposed approach considers the relative importance of criteria(3)
The proposed approach considers the subjective preferences of experts(4)
The traditional weighted arithmetic average approach and DEA method can be viewed as special cases of the proposed approach(5)
The proposed approach provides a more flexible contractor selection technique to support contractor assessment for selectionThis paper assumed that the experts have the same weight and did not consider the objective weight of the evaluation data. Further research studies can explore the simultaneous considerations of subjective and objective weights of assessment criteria to handle different field decision-making issues.



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*Source: 3664709-2021-11-18.xml*