Multi-Criteria Decision Making of Contractor Selection in Mass Rapid Transit Station Development Using Bayesian Fuzzy Prospect Model

: In Taiwan, the most advantageous tender in governmental procurement is the selection of a general contractor based on a score or ranking evaluated by a committee. Due to personal, subjective preferences, the contractor selection of committee members may be di ﬀ erent, causing cognitive di ﬀ erence between the results of the members’ selection and the preliminary opinions provided by the working group. Integrated, multi-criteria decision making techniques, combined with preference relation, Bayesian, fuzzy utility, and prospect theories are used to assess factors weighing up the duration / cost / quality, probability of external information, and utility function system. The paper proposes a Bayesian fuzzy prospect model for group decision making, based on probability and utility multiplied relation, and taking the sustainable development factors into consideration. This study aims to provide committees with an objective model to select the best contractor for public construction projects. The results of this study can avoid the lowest bidder being selected; besides, the score gap of contractor selection can be increased, and the di ﬀ erence between the top three contractors’ scores can be decreased as well. In addition to proposing an innovative decision-making system of contractor selection and an index weight-assessing system for sustainable development, this model will be widely applied and sustainably updated for other cases.


Introduction
Multiple criteria decision making (MCDM) is considered a complex decision-making tool involving both quantitative and qualitative factors. In recent years, several MCDM techniques and approaches have been suggested to choose the optimal probable options [1]. Such applications have been widely investigated in both the theory and practice of MCDM [2]. In the present study, a MCDM method was developed as follows. First, factors pertaining to duration, cost, and quality were determined. These factors are influential factors in MCDM [3,4]. Subsequently, the fuzzy preference relation (FPR) was adopted to construct a paired decision-making matrix of preferences [5]. This method enables decision makers to express preferences regarding a set of alternatives using the least number of judgments; the method also makes it unnecessary to examine the consistency of the decision-making process [6]. A fuzzy analytical hierarchy process (FAHP) method was used to prioritize the identified risks [7][8][9], and Fuzzy-TOPSIS achieved through the application of order preference by Similarity to Ideal Solution and fuzzy sets theory [10,11]. Second, Bayes' theorem (BT) was used as it provides a natural theoretical framework for explicitly articulating epistemic or state-of-knowledge uncertainties OCRA and Grey (ranking order) (investment in the most appropriate type of hotels) 7 Khanzadi, Turskis, Amiri and Chalekaee [29] 2017 Game theory, ADR, grey number (solve dispute resolution problems in construction) 8 Mokhtariani, Sebt and Davoudpour [33] 2017 Cultural heritage Building renovation -Construction marketing -Attribute Analysis: Service attributes versus construction 9 Pashaei and Moghadam [9] 2018 Fuzzy AHP Method (Alternative in Low-Rise Buildings) 10 Ilce and Ozkaya [25] 2018 AHP and MOORA methods (the raised floor choice practice consists) 11 Mardani, Jusoh, Halicka, Ejdys, Magruk and Ahmad [19] 2018 A review MCDM -MOORA, COPRAS, ARAS, WASPAS, SWARA -classified into 10 areas: (1) energy source, (2) buildings, (3) material, (4) project management, (5) construction management, . . . 12 Hasnain, Thaheem and Ullah [27] 2018 ANP-Based Decision Support System -Analytical network process (ANP) (Contractor Selection in Road Construction) 13 Liang, Zhang, Wu, Sheng and Wang, [21] 2018 Using Competitive and Collaborative Criteria with Uncertainty (Joint-Venture Contractor Selection) 14 Alpay and Iphar [11] 2018 fuzzy multi-criteria decision-making methods -Fuzzy TOPSIS and fuzzy VIKOR (Equipment selection)  16 Ye, Zeng and Wong [37] 2018 Competition rule of the multi-criteria approach -34 tender evaluation factors are proposed to compose the competition rule in China -The composition varies slightly between public and private sectors 17 Ortiz, Pellicer and Molenaar [38] 2018 Management of time and cost contingencies in construction projects: a contractor perspective (a case study of two large Spanish construction companies) 18 Cao, Esangbedo, Bai and Esangbed [30] 2019 Contractor Selection MCDM Problem Grey -SWARA-FUCOM Weighting Method (Floating Solar Panel Energy System Installation) 19 Turskis, Goranin, Nurusheva and Boranbayev [20] 2019 Fuzzy WASPAS and AHP methods (Determine Critical Information Infrastructures of EU Sustainable Development) 20 Antoniou and Aretoulis [18] 2019 TOPSIS and utility theory (highway construction contractors) 21 Morkunaite, Bausys and Zavadskas [26] 2019 WASPAS-SVNS Method (Contractor Selection for Sgraffito Decoration of Cultural Heritage Buildings) 22 Gunduz and Alfar [32] 2019 AHP Method (Innovation in project management) 23 Morkunaite, Podvezko, Zavadskas and Bausys [31] 2019 AHP, PROMETHEE(Ranking) (Contractor selection by Cultural heritage buildings) 24 Davoudabadi, Mousavi, Shaparauskas and Gitinavard [23] 2019 a new uncertain weighting and ranking based on compromise solution with linear assignment approach -Interval-valued intuitionistic fuzzy sets (IVIFSs) -Ranking (in energy projects-A case study about the construction project selection problem) 25 Aladag and Isik [7] 2020 Fuzzy AHP Method (BOT project-A case study of a PPP airport project) 26 Mahamadu, Manu, Mahdjoubi, Booth, Aigbavboa and Abanda [10] 2020 Fuzzy TOPSIS (BIM capability assessment: Post-selection performance of organizations on construction projects) 27 Koc and Gurgun [35] 2020 AH P, MCDM (Contractor prequalification for green buildings-Evidence from Turkey) 28 Zhang [36] 2020 AHP, D-S Evidence Theory (Construction in Government public project green procurement in China)

Preference Relationships Theory
Preference Relationships Theory (PRT) was adopted to calculate the relative weights between factors. The decision-making process is largely based on the preference relation for alternatives. The preference relation is a value assigned by experts to two alternatives to reflect the experts' preferences for the two alternatives. Preference relations can be applied in a decision-making model to integrate experts' individual preferences into a group preference [39][40][41][42][43]. In decision making, two types of preference relations are adopted: multiplicative preference relation (MPR) and FPR [39,44].
The advantages of the combination between MPR and FPR were to develop a possibility evaluation method. First, MPR and FPR matrices were used to define linguistic variables and quantized values corresponding to linguistic variables. Subsequently, a questionnaire was administered to collect the subjective opinions of each evaluator. To integrate the experts' opinions and obtain the implementation possibility, the questionnaire results were then converted to the FPR's average weight method.

Bayes' Theorem
Bayes' theorem (BT) is presented in Equation (2), where the conditional probability theorem is for before or after an event [45,46].
where A and B are events and p(B) > 0; p(A|B) is the probability of event A occurring if event B occurs; p(B|A) is the probability of event B occurring if event A occurs; p(A): prior probability density function; p(B): prior probability density function (or marginal probability function), which indicates the probability of X occurring in a sample dataset; p(B|A): Likelihood function, sample distribution; p(A|B): Posterior probability density function.

1.
The prior probability is expressed as a cumulative distribution function (CDF) as follows: where w 1 (p) is prior probability density function; α and β are the parameters of Equation (2).

2.
The maximum likelihood distribution depends on additional information.

4.
Posterior probability = prior probability × likelihood function. The posterior probability is expressed as a probability density function (PDF) as follows:

Prospect Theory
Expected utility theory postulates that all possible outcomes of an uncertain event have their respective utilities and probabilities, where the sum of the utility-probability product of each possible outcome represents the expected utility of an uncertain event. Subsequently, PT adopts probability weighting functions to explain the non-linear preferences of people when they evaluate the probability of uncertain events [47] (pp. 282-283). Specifically, every result corresponds to the product of decision weights (subjective probability) and psychological values (utility), and this product represents the decision prospect value.

1.
PT and investment psychology (a four-level model): Kahneman and Smith [13] proposed the S-shaped utility function in Foundations of Behavioral and Experimental Economics. It has experimentally indicated that people have non-linear preferences when evaluating probabilities [35]. This preference is characterized by (1) a tendency for "loss aversion," in which a unit loss is perceived to be of a greater magnitude than a unit gain; (2) a tendency for "risk aversion" in gain situations; (3) a tendency for "risk seeking" in loss situations; and (4) a tendency to make decisions based on a "reference point" to determine gain or loss situation. CPT: This theory uses a cumulative probability to convert an expected utility probability. Tversky and Kahneman [14,48] and Prelec [49] have proposed different parameters for the probability weighting function. 3.
CCPT: Ali and Dhami [15] (pp. 14-16) proposed a probability weighting function which uses the composite Prelec probability weighting function (CPF) [49] to correct the curve function of high-and low-probability zones. Due to this correction, changes in subjective decision-making probabilities after the provision of external information can be better reflected.

Influence Factors Considered
In the European 2020 strategy, the EU mentioned three trends that strengthen economic and social development: "smart growth", "sustainable growth", and "inclusive growth". For the sustainable growth of the construction industry, Taiwan must meet the "sustainable public engineering indicators" issued by the Public Works Commission: safety, creativity, humanities, durability, waste reduction, energy saving, ecology and benefits.

Summary
According to the literature review above, the merits of PRT, BT, FUT, and PT were combined to develop BFPM for the use in a contractor selection MCDM. Utility theory and fuzzy statistical methods in PRT were used to model the uncertainties of qualitative factors and the risk preferences of subjective utilities. The use of PRT addressed consistency-related issues in the pairwise comparison matrix of the AHP method. PRT, BT, and FUT were integrated to represent expert knowledge on state evaluations and to model the expected values that can be acquired using the product of probability and utility. BT, an effective method for assessing the posterior probabilities of the influential factors once additional information was provided, was applied to simulate the probability of success, which was then used to determine the probability that bid commitment was implemented. The strengths of BFPM are to help owner to make the optimal contractor selection decision with taking not only the utilities of bid contractors but also the probability of bid implementation into account. Therefore, BFPM was used to model the transformation of risk or uncertainty in contractor selection, capturing the difference relationship between MCDM and BFPM. A public project includes a concept of life cycle risk management, and only after facilities are smoothly and successfully completed, they can be operated and maintained. The entire project life cycle incorporates four main stages, including planning and design, project bidding, project performance, and operation and maintenance, as well as each stage carefully selecting contractors to handle the corresponding tasks. When each stage has selected its contractor, through the series of works in each stage, government can ensure the quality of the public buildings or facilities, reduce maintenance costs, increase facility efficiency, and control energy consumption.

Constructing the Bayesian Fuzzy Prospect Model
This paper presents a decision-making procedure for selecting a general contractor for construction projects. Moreover, this study examined the duration discount, cost discount, and quality assurance, which are the three influence factors belonging to the sustainable development indicators in a contractor's bid commitment. The duration discount, d, is defined as the proportion of the difference between committee members' expected duration and the bid duration to committee members' budgeted duration. The cost duration, c, is defined as the proportion of the difference between committee members' expected cost and the bid cost to the owner's budgeted cost. The quality assurance, q, is defined as the proportion of the difference between the bid commitment and committee members' or owners' quality requirement. Each commitment is assumed to have two possible outcomes for the contract performance after a bid is won: success or failure (in the implementation). Probability theory assumes that these two possible outcomes can occur. The terms (pd S , pd F ), (pc S , pc F ), and (pq S , pq F ) represent the possibilities of the two outcomes of bid commitment in terms of the duration discount, cost discount, and quality assurance, respectively, and are collectively referred to as "the implementation probability of bid commitment." The case example of the present study is a construction project involving mass rapid transit station development. Through the Bayesian probability evaluation method, the implementation possibility for the case study was obtained. The duration of the project was 40 months, and the budgeted cost was 199 million Taiwan Dollars (TWD).
The FPR matrix was adopted to construct an evaluation model. This model enabled committee members to identify the relative importance of factors and to evaluate the implementation possibility of bid commitment. Subsequently, the evaluation model was integrated with PT to develop a decision-making model. The Bayesian probability model and fuzzy PT comprised four parts. The stage for constructing the Bayesian fuzzy prospect model (BFPM) is shown in Figure 1. The FPR matrix was adopted to construct an evaluation model. This model enabled committee members to identify the relative importance of factors and to evaluate the implementation possibility of bid commitment. Subsequently, the evaluation model was integrated with PT to develop a decision-making model. The Bayesian probability model and fuzzy PT comprised four parts. The stage for constructing the Bayesian fuzzy prospect model (BFPM) is shown in Figure 1.

Asssessment for Implementation Possibility
Preference relationship theory was used to (1) evaluate the influential factors for selecting a contractor, (2) determine the relative weights between influential factors, and (3) evaluate the implementation possibility of bid commitment. First, the duration, cost, and quality factors were determined through a literature review and organized into factors recommended for use in this study. Subsequently, the MPR and FPR were used to calculate the weights of the duration, cost, and quality factors. The MPR and FPR were also employed to evaluate the implementation possibility of bid commitment according to two possible outcomes: success and failure (in the implementation).

Derivation of Expected Probability
BT was used to determine (1) the prior probability weighting function and (2) the Bayesian probability weighting function. Due to external environmental information, the prior probability weighting function was obtained by using the CPT probability function and the parameter value. The posterior probability, which was based on the CCPT, was used to derive a likelihood function that satisfied Bayes' theorem. This function was used as the expected probability of a contractor being selected by committee members.

Evaluation of Utilities for Duration, Cost and Quality
FU theory was used to (1) determine the fuzzy utility function (FUF) and (2) evaluate the utility of bid commitment for the committee members. To determine the FUF, we adopted the FUF proposed by Kirkwood [16] and referred to the utility function provided by Cheng and Kang [17]. Expert questionnaires were collected and organized, and the FUF was established. Subsequently, the center-of-gravity method was employed to evaluate the differences in utility between potential contractors. After the duration discount, cost discount, and quality assurance (%) were converted, the FUF was used to calculate the utility of bid commitment for the committee members.

Overall Prospect Evaluation of Candidate Contractors
CCPT was used to (1) calculate the posterior probability of bid contractors, (2) evaluate the overall prospect value of bid contractors, and (3) select the optimal contractor. The posterior probability of bid contractors was calculated by evaluating the relevant expected probability of committee members and then multiplying it with the utility of bid commitment to obtain the overall prospect value of a potential contractor for reference. After the contractors were ranked, the optimal and runner-up applicants were selected. Finally, the contractor selection results obtained from BFPM were compared with the lowest tender and MCDM (overall utility values [4]), as well as the multi-criteria prospect model (MCPM) results.

Identifying Influence Factors of Duration, Cost and Quality Implementation
Sixteen influence factors were selected, of which, seven (DF k , k = 1, 2, . . . , 7) were used for the duration discount, four (CF k , k = 1, 2, 3, 4) were used in cost discount implementation, and the remaining five (QF k , k = 1, 2, 3, 4, 5) were used in quality assurance implementation. Among them, 5 influence factors belong to sustainable growth indexes, including DF1, DF4, QF1, QF2 and QF3. Technical ability (DF1) refers to creativity; plan management (DF4) refers to safety, health, and environment protection; building materials (capacity)/equipment resources (QF1) refers to green building mark; after sales service (QF2) refers to the feedback facility about humanities; and warranty period (QF3) refers to waste reduction and energy saving. QF2 and QF3 of influence factors can achieve sustainability in the operation stage of construction life cycle (see Table 2 for details).

Determining Relative Weights between Influence Factors
The MPR of an alternative set X can be expressed using matrix A = (aij), where (aij) represents the intensity of a preference for alternative xi relative to alternative xj. In other words, the intensity of a preference for alternative xi is aij times that for alternative xj. In addition, the diagonal matrix of the MPR matrix has a multiplicative reciprocal relationship. According to Satty, the multiplicative reciprocal MPR matrix A = (aij) must be consistent, that is, aij · ajk = aik, . . . , ∀i, j, k ∈ {1, . . . , n}. Table 3 presents the linguistic variables defined according to fuzzy theory and MPR. These variables can be used by evaluators or decision makers to indicate the relative importance of each factor and their preference for the possibility of duration/cost/quality (D/C/Q) commitment being implemented. This study defined simple linguistic terms quantified on the scale [1/5, 1/4, 1/3, 1/2, 1, 2, 3, 4, 5], with each item symbolized by VL, LAL, L, FL, F, FH, H, HVH, and VH, respectively. This scale allows evaluators to express the relative degree of importance and implementation probability of the D/C/Q commitment.
For example, if the importance of DF 1 relative to DF 2 is "very high," then DF 1 is five times more important than DF 2 is. By using the same set of linguistic variables, the possibility of "success implementation" for "failure implementation" (represented by S/F) and the possibility of "failure implementation" for "success implementation" (represented by F/S) are evaluated according to the bidder's duration, cost, and quality commitments. Details are presented in Section 4.3.
x times on the seven factors that influence the implementation of the duration discount DF k , k = 1, 2, . . . , 7. A questionnaire, answered by six experts with experience in contractor selection, was then used to evaluate the relative importance of each factor. We have drawn up the statement, and the experts agreed to adopt anonymity and make the questionnaire contents public in the beginning of questionnaire. The questionnaire results of Evaluator No.1 are presented in Figure 2 as an example.
Sustainability 2020, 4, x FOR PEER REVIEW 10 of 34 experts agreed to adopt anonymity and make the questionnaire contents public in the beginning of questionnaire. The questionnaire results of Evaluator No.1 are presented in Figure 2 as an example. In Figure 2, H indicates that DF1 is more important than DF2 (specifically three times more important than DF2, as stated in Table 3). The evaluator's preference corresponding to the six selected symbols based on the definition of linguistic variables (Table 3) The MPR matrix was constructed through four steps. The matrix was used to analyze the relative importance between the factors that influence the implementation of the duration discount {DF1, …, DF7}. The steps in the matrix construction were as follows: Step 1. Use Equation (4) For example, a(df)17 [17].
Step 2: Construct the following MPR matrix.
Step 3: Identify the maximum value in Matrix A: Step 4: Convert Matrix A into a consistent MPR matrix = ( ), as presented in Equation (7).
The FPR of alternative set X can be expressed using matrix P = [pij], where [pij] represents the intensity of preferences for alternative xi relative to alternative xj, and pi is between 0 and 1, as determined by the fuzzy membership function. A pij value of 0.5 indicates indifference between xi and xj; pij = 1 indicates that xi is highly preferred over xj; pij = 0 indicates that xj is highly preferred over xi. In addition, the diagonal matrix of the FPR matrix is assumed to be an additive reciprocal matrix, that is, pij + pji = 1, ∀i, j ∈ {1, …, n}. In Figure 2, H indicates that DF 1 is more important than DF 2 (specifically three times more important than DF 2 , as stated in Table 3). The evaluator's preference corresponding to the six selected symbols based on the definition of linguistic variables (Table 3) was {3, 2, 1, 1, 2, 3}. These six preference values are expressed using the corresponding set of variables {a(df ) 12 , a(df ) 23 , a(df ) 34 , a(df ) 45 , a(df ) 56 , a(df ) 67 }.
The MPR matrix was constructed through four steps. The matrix was used to analyze the relative importance between the factors that influence the implementation of the duration discount {DF 1 , . . . , DF 7 }. The steps in the matrix construction were as follows: Step 1. Use Equation (4) to calculate all preference values for set B (the set of preference values).
For example, a(df Step 2: Construct the following MPR matrix.
where A is the MPR for the relative weights of df k; Step 3: Identify the maximum value in Matrix A: Step 4: Convert Matrix A into a consistent MPR matrix C = f (A), as presented in Equation (7).
The FPR of alternative set X can be expressed using matrix P = [pij], where [pij] represents the intensity of preferences for alternative xi relative to alternative xj, and pi is between 0 and 1, as determined by the fuzzy membership function. A pij value of 0.5 indicates indifference between xi and xj; pij = 1 indicates that xi is highly preferred over xj; pij = 0 indicates that xj is highly preferred over xi. In addition, the diagonal matrix of the FPR matrix is assumed to be an additive reciprocal matrix, that is, pij According to the definition of the consistent FPR matrix [61], the domain range was [1/5, 5]. Moreover, the consistent MPR matrix C can be converted into an FPR matrix with a domain range of By using Equation (8), the six evaluators' FPR matrices (D 1 , D 2 , . . . , D 6 ) were converted, and the average FPR matrix E is given as follows: The average FPR matrix E = [e(df ) ij ] was calculated to obtain the normalized FPR matrix Q = [q(df ) ij ], which is expressed as follows.
The FPR matrix is presented in Table 4, where the relative weights r(df )i for the seven influence factors of duration discount were obtained through the following Equation (11) and the relative weights r(df )i for the four influence factors of duration discount were calculated as 0.18, 0.13, 0.11, 0.15, 0.16, 0.16 and 0.11 in Table 4. By using the aforementioned steps, the relative weights r(cf ) i for the four influence factors of cost discount were calculated as 0.30, 0.24, 0.23, and 0.23. Subsequently, the relative weights r(qf ) i for the five influence factors of quality assurance were calculated as 0.27, 0.17, 0.14, 0.20, and 0.22.

Assessing the Probability of Fulfilling the Bid Commitment
With reference to the prediction results of Wang and Chang [62] (pp. 807-810) on the possibility of successful knowledge management implementation, we formulated the following steps to evaluate the implementation possibility of the D/C/Q commitments of a bid.
Step 1: Define linguistic variables and design the questionnaire.
Every candidate contractor was required to provide evidence relating to factors affecting DF k , CF k , and QF k implementation. Then, with respect to DF k , evaluators El to E6 were asked to select one preference linguistic term for success over failure.
Step 2: The evaluators completed the questionnaire by referring to the evidence on each factor. The implementation possibility of contractor A's duration commitment was evaluated. Table 5 presents the six evaluators' questionnaire results.
For each influence factor, the evaluators performed a relative comparison of S/F and obtained quantized values according to the definition of linguistic variables.
Step 3: Convert quantized values into FPR values. The MPR value ads kl ∈ 1 5 , 5 was converted into a consistent FPR value bds kl ∈ [0, 1]. The terms bds kl = 1 2 (1 + log 5 ads kl ) and cds k = 1 m m l=1 bds kl were calculated, where cds k is the preference rating success/failure related to DF k.
Step 4: Determine the group fuzzy preference value. The mean quantized value of the six evaluators was expressed as cds k , where k = 1, . . . , 7. This value can be used to evaluate the implementation possibility of duration commitment ( Table 6). The analogous mean values ccs k , where k = 1, . . . , 4, and cqs k , where k = 1, . . . , 5, can be used to evaluate the implementation possibilities of cost commitment and quality commitment, respectively. Step 5: Normalize a series of FPRs to calculate the assessment values corresponding to each factor. Each consistent FPR matrix was normalized using the Equation (12) to derive the normalized FPR matrix Q k .
Step 6: Combine the weights of the influence factor to determine the implementation possibility. The normalized FPR matrix and Equation (13) can be used in the analysis to obtain a set of two degrees of implementation [rd 1 , rd 2 ] (Table 7). Corresponding to the seven influence factors of duration discount DF k , where k = 1, 2, . . . , 7, the seven sets of implementation degrees were grouped to form a possibility analysis matrix [r(df ) k ]. Subsequently, the relative importance matrix [r(df ) k ] of the seven factors DF k,, where k = 1, 2, . . . , 7, which was obtained using Equation (8), was calculated to determine the possibilities of the two outcomes of duration discount as follows: where pdi is the probability on implementation of i's time.
The successful implementation probability of contractor A's duration discount was evaluated. Table 8 presents the analysis results. To evaluate the successful implementation probabilities for cost discount and quality assurance, the steps mentioned above were repeated. The results are presented in Table 7. In addition, the relative weights between each influence factor obtained in Section 4.2 were multiplied. The results indicated that contractor A had success implementation probabilities of 0.76, 0.68, and 0.72 for D/C/Q commitments, respectively. The aforementioned method can be used to determine the successful implementation probabilities of the D/C/Q commitments of other contractors.

Assessment of the Committee Members' Expected Probability
By using BT, the assessment group first determined the prior and posterior cumulative probability weight function. Among these functions, the prior probability weight function is the CPT probability function and parameter value. Because of external environmental information, the posterior probability was based on CCPT. BT was used to convert the CDF into a PDF to derive the likelihood function L(p). This function was a normal distribution, which indicated that the CCPT is in agreement with BT. In addition, the result can be regarded as the expected probability that committee members will select a contractor. The derivation process of the Bayesian probability weighting function is illustrated in Figure 3.

Determination of the Prior Probability Weight Function
This research used CPT probability weighting function [45] as the prior probability weight function (see Figure 4). Subsequently, according to the parameter results calculated by Cheng and Kang through a questionnaire, namely α = 0.62 and β = 0.97 [17] (p. 1059), the prior probability function was obtained using the following equation:

Determination of the Prior Probability Weight Function
This research used CPT probability weighting function [45] as the prior probability weight function (see Figure 4). Subsequently, according to the parameter results calculated by Cheng and Kang through

Derivation of the Bayesian Probability Weight Function
CCPT modifies the curves of low probability and high probability in CPT [15]. The Prelec function, which is presented in the middle part of Figure 5, is usually not conformed to in BT and other relevant theories of probability in which uncertain results are expected [9]. As the decision maker considers tenders with high probability as priority contractors, the present research focused on the high-probability zone (p = 0.66-1). The posterior probability of BT was set as the risk probability ( Figure 5).

Derivation of the Bayesian Probability Weight Function
CCPT modifies the curves of low probability and high probability in CPT [15]. The Prelec function, which is presented in the middle part of Figure 5, is usually not conformed to in BT and other relevant theories of probability in which uncertain results are expected [9]. As the decision maker considers tenders with high probability as priority contractors, the present research focused on the high-probability zone (p = 0.66-1). The posterior probability of BT was set as the risk probability ( Figure 5).

Derivation of the Bayesian Probability Weight Function
CCPT modifies the curves of low probability and high probability in CPT [15]. The Prelec function, which is presented in the middle part of Figure 5, is usually not conformed to in BT and other relevant theories of probability in which uncertain results are expected [9]. As the decision maker considers tenders with high probability as priority contractors, the present research focused on the high-probability zone (p = 0.66-1). The posterior probability of BT was set as the risk probability ( Figure 5).
Take exp on both sides;

2.
Similarly, obtain p = e −(  Step 1: Define the high-probability zone (p = 0.66-1) and use the conditional probability relation of Bayes' theorem to calculate the Bayesian relation of two connected probabilities. Calculate the Bayesian relation of two connected probabilities and subsequently derive the Bayesian probability distribution through the steps in Table A2.
Step 2: After dividing the Bayesian relations of the preceding and following terms, let L(p 1 ) = 1 and sum the overall Bayesian relations' values.
Step 3: First, let L(p 1 ) = 1, obtain L(p 2 )-L(p n ) form Equations (17a)-(17e) and sum the values of the overall Bayesian relation. Subsequently, divide the value of n i=1 L(p i ) into each formula to obtain the likelihood function L(p i ) and calculate the Bayesian probability while satisfying the equation n i=1 L(p i ) = 1.
Step 4: Using the relation between the high-probability zone (P = 0.66-1) and the conditional probability in Bayes' theorem, obtain the Bayesian probability by summing and weighting. As illustrated in the left-hand-side picture in Figure 6, the CDF of the Bayesian probability distribution (L(p i ) = −0.14 × (p i − 0.85) × 2 + 0.07) approximates a normal distribution (R 2 = 0.9548) with a peak at 0.85.
Step 5: As illustrated in the right-hand-side picture in Figure 6, the PDF of the Step 5: As illustrated in the right-hand-side picture in Figure 6, the PDF of the Bayesian probability ( (P ) = −0.21 × ( − 0.51) × 2 + 0.05) distribution also approximates the normal distribution (R 2 = 0.9999).
In summary, the high-probability zone of the CCPT approximates the posterior probability of BT. In other words, the provision of external information helps committee members to increase their subjective risk probability when selecting contractors.

Assessment of the Utility of Bid Commitment
The present research used the FUF developed by Kirkwood [16] and Cheng and Kang [17], which is an extension of this approach, to incorporate the uncertainties of the expert estimates. Expert questionnaire results were collected and FUFs for D/C/Q commitments were established. Subsequently, the utility of bid commitment for committee members was evaluated by defuzzifying and defining the D/C/Q fuzzy intervals. Thus, the difference between the utilities of influence factors for candidate contractors and the D/C/Q fuzzy weights were determined. Figure 7 illustrates In summary, the high-probability zone of the CCPT approximates the posterior probability of BT. In other words, the provision of external information helps committee members to increase their subjective risk probability when selecting contractors.

Assessment of the Utility of Bid Commitment
The present research used the FUF developed by Kirkwood [16] and Cheng and Kang [17], which is an extension of this approach, to incorporate the uncertainties of the expert estimates.
Expert questionnaire results were collected and FUFs for D/C/Q commitments were established. Subsequently, the utility of bid commitment for committee members was evaluated by defuzzifying and defining the D/C/Q fuzzy intervals. Thus, the difference between the utilities of influence factors for candidate contractors and the D/C/Q fuzzy weights were determined. Figure 7 illustrates the workflow for evaluating the utility of bid commitment for committee members. The center-of-gravity method was employed to evaluate the difference between the utilities of candidate contractors. After the duration discount, cost discount, and quality assurance data (in %) were converted, the FUF was used to calculate the utility of the bid commitment for committee members.

Determining the Fuzzy Utility Functions
The exponential utility function ( ) was adopted. In the expression for ( ), x represents the preference in decision making, is the risk tolerance of a decision maker, "Low" represents the least preferred candidate, "High" represents the most preferred candidate, and reflects the personality of a decision maker (conservative, neutral, or adventurous) [16] (p. 6). (1) ≥ 0 Conservative (risk-averse nature) (2) < 0 Adventurous (risk-seeking nature)

Determining the Fuzzy Utility Functions
The exponential utility function ( ) was adopted. In the expression for ( ), x represents the preference in decision making, is the risk tolerance of a decision maker, "Low" represents the least preferred candidate, "High" represents the most preferred candidate, and reflects the personality of a decision maker (conservative, neutral, or adventurous) [16] (p. 6).

Determining the Fuzzy Utility Functions
The exponential utility function (x) was adopted. In the expression for µ(x), x represents the preference in decision making, ρ is the risk tolerance of a decision maker, "Low" represents the least preferred candidate, "High" represents the most preferred candidate, and ρ reflects the personality of a decision maker (conservative, neutral, or adventurous) [16] (p. 6).
(1) ρ ≥ 0 Conservative (risk-averse nature) (2) ρ < 0 Adventurous (risk-seeking nature) The FUF and shapes are defined Equation (18) as follows: Specific utility functions, which were established using the data obtained through an expert questionnaire [12] (p. 1057) and the aforementioned information, were used to model the personality of a decision maker (ρ). Duration utility function:

Evaluating a Committee Members' Utility of the Bid Commitment
Three defuzzification methods exist: the center-of-gravity, center of maxima, and center of sums methods. In this study, the center-of-gravity method was adopted to determine the difference degrees of the D/C/Q fuzzy utilities of the candidate contractors. The center of sums method was used to calculate the D/C/Q fuzzy weights to forecast the utility of bid commitment for committee members.

1.
Solve fuzzy weights relative to the D/C/Q factors The center of sums method was adopted to determine the fuzzy weights between the D/C/Q factors by conducting group decision analysis among the evaluators (Table 9). Subsequently, Equations (19a)-(19g) were used to calculate the ratios of project duration to the cost and the quality fuzzy weights. Equations (19d)-(19f) were used to determine the ranking function of the triangular fuzzy number [63]. The result was wd:wc:wq = 0.402:0.302:0.296. Thus, from each committee members' perspective, the completion of a project within the assigned duration is the most crucial aspect, followed by meeting the cost and quality requirements. The cost and quality requirements were of equal importance. where wd, wc, wq are weight of the D/C/Q factors; U W 1j , U W 2 j , U W 3 j are the triangular fuzzy number of the D/C/Q factors.

2.
The steps used to determine the difference between the candidate contractors with respect to the D/C/Q fuzzy utility are as follows: Step 1: Establish a membership function for linguistic variables pertaining to the aforementioned difference.
Fuzzy statistical analysis [64] (pp. 71-72) was used to establish fuzzy ratings for linguistic variables, where the variables were rated in terms of the five levels: very high (VH), high (H), indifference (I), low (L), and very low (VL). Subsequently, fuzzy additive and scalar multiplication methods were used to analyze evaluator responses to questionnaires on the differences for linguistic variables [65] (pp. 231-232). Specifically, the fuzzy numbers representing such difference were calculated according to the method of Cheng and Hsiang [66] to obtain the membership function (Table 10 and Figure 9). Finally, based on the relative frequency (degree of membership function), the statistical linguistic variables were grouped to obtain and correct the fuzzy number of the membership functions.
Step 2: Evaluation of the degree of difference. The linguistic variables used by evaluators were analyzed to assess the difference between the duration discounts of contractors A and B. 1 n (A 1 ⊕ · · · ⊕ A n ) = 1 n × (a 1 + · · · + a n ), · · · , 1 n × (l 1 + · · · + l n ) n = 5 (20) where A 1 -A n are fuzzy numbers for experts to assess the degree of difference; a 1 -a n , . . . , l 1 -l n are the fuzzy number of linguistic variables; n is number of experts. variables, where the variables were rated in terms of the five levels: very high (VH), high (H), indifference (I), low (L), and very low (VL). Subsequently, fuzzy additive and scalar multiplication methods were used to analyze evaluator responses to questionnaires on the differences for linguistic variables [65] (pp. 231-232). Specifically, the fuzzy numbers representing such difference were calculated according to the method of Cheng and Hsiang [66] to obtain the membership function (Table 10 and Figure 9). Finally, based on the relative frequency (degree of membership function), the statistical linguistic variables were grouped to obtain and correct the fuzzy number of the membership functions.
(a) (b) The experts filled in a form of linguistic variables, which are VH, L, VH, I, and H in Table 10. Equation (20) was used to obtain the fuzzy numbers representing the evaluation results of five experts. Specifically, the numbers pertained to the difference between the bidding prices of contractors A and B. Through the calculation [0.2 × (6.50 + 1.50 + 6.50 + 3.50 + 5.50), . . . , 0.2 × (10.00 + 6.50 + 10.00 + 8.50 + 9.50)], the comprehensive fuzzy values were determined to be 4.70, 5.27, 6.02, 6. 33, 7.10, 7.30, 7.82, 7.91, 8.07, 8.44, 8.53, and 8.90. The details of the calculations are presented in Figure 10. The center-of-gravity method was used to defuzzify the integrated fuzzy numbers. Using a difference score (x) of 6.9571 and a scale of −10 to 10 (which represents a difference of −20% to 20%), a difference ratio of 7.83% was obtained through linear conversion. The degrees of difference in the utilities of duration discount, cost discount, and quality assurance between contractor A and the other contractors were estimated (Table 11).  Step 2: Evaluation of the degree of difference. The linguistic variables used by evaluators were analyzed to assess the difference between the duration discounts of contractors A and B.
where A1-An are fuzzy numbers for experts to assess the degree of difference; a1-an, …, l1-ln are the fuzzy number of linguistic variables; n is number of experts. The experts filled in a form of linguistic variables, which are VH, L, VH, I, and H in Table 10. Equation (20) was used to obtain the fuzzy numbers representing the evaluation results of five experts. Specifically, the numbers pertained to the difference between the bidding prices of contractors A and B. Through the calculation [0.2 × (6.50 + 1.50 + 6.50 + 3.50 + 5.50), …, 0.2 × (10.00 + 6.50 + 10.00 + 8.50 + 9.50)], the comprehensive fuzzy values were determined to be 4.70, 5.27, 6.02, 6. 33, 7.10, 7.30, 7.82, 7.91, 8.07, 8.44, 8.53, and 8.90. The details of the calculations are presented in Figure 10. The center-of-gravity method was used to defuzzify the integrated fuzzy numbers. Using a difference score (x) of 6.9571 and a scale of −10 to 10 (which represents a difference of −20% to 20%), a difference ratio of 7.83% was obtained through linear conversion. The degrees of difference in the utilities of duration discount, cost discount, and quality assurance between contractor A and the other contractors were estimated (Table 11).   Step 3: Present the evaluation results of the utility of the bid commitment for the committee members. The expert questionnaire scores were averaged to determine the duration discount, cost discount, and quality assurance values of contractor A. Subsequently, the difference in utility between each contractor (Table 11) was converted to calculate the duration discount, cost discount, and quality assurance values. The contractor's D/C/Q utilities were also calculated using the data presented in Figure 10. Finally, the ratios of the D/C/Q fuzzy weights, that is, wd:wc:wq = 0.402:0.302:0.296, were multiplied to obtain the overall utility values, which are listed in Table 12.

Evaluation of Overall Prospect Value of Candidate Contractors
This section focuses on CCPT. The posterior probability of candidate contractors (as calculated in Sections 4.3 and 5.2) were multiplied with the utility of bid commitment for committee members (presented in Section 6.2) to obtain the overall prospect value of the candidate contractors. Figure 11 illustrates the evaluation steps. After the contractors were ranked, the best and second-best contractors were selected. Finally, the contractor selection results obtained from BFPM were compared with the lowest tender and overall utility values as well as the MCPM results.

Calculation of the Posterior Probability of Candidate Contractors
The implementation possibilities for the bidder's duration, cost, and quality commitments (probability of success) presented in Section 4.3 were converted to the prior probability of duration, cost, and quality factors for committee members through the Bayesian probability weighting functions (Section 5.1). The aforementioned possibilities were also converted to the posterior probability through the Bayesian probability weighting functions (Section 5.2). Subsequently, the posterior probabilities of candidate contractors were summarized to determine the probability of the subjective perception (of risk) of committee members towards a bid contractor. Table 11 lists the prior and posterior probability results.

Figure 11.
Steps for evaluating the overall prospect value of candidate contractors.

Calculation of the Posterior Probability of Candidate Contractors
The implementation possibilities for the bidder's duration, cost, and quality commitments (probability of success) presented in Section 4.3 were converted to the prior probability of duration, cost, and quality factors for committee members through the Bayesian probability weighting functions (Section 5.1). The aforementioned possibilities were also converted to the posterior probability through the Bayesian probability weighting functions (Section 5.2). Subsequently, the posterior probabilities of candidate contractors were summarized to determine the probability of the subjective perception (of risk) of committee members towards a bid contractor. Table 11 lists the prior and posterior probability results.

Evaluation of the Overall Prospect Value of Candidate Contractors
The overall prospect value of a candidate contractor, in terms of the duration, cost, and quality commitments (Table 13), was obtained by multiplying the results listed in Table 10 (utility of bid commitment for committee members) with the results presented in Table 13 (the posterior probability of candidate contractors for committee members). As listed in Table 14, contractor C had the highest prospect for duration, followed by contractors E and A; contractor D had the highest prospect for cost, followed by contractors E and A; and contractor A had the highest prospect for quality, followed by contractors C, D, and E.  The D/C/Q analysis revealed that contractor D provided the largest duration discount. However, as the evaluators perceived contractor D to be least likely to implement its commitments, contractor D had the smallest expected utility value. The opposite was the case for contractor C. Specifically, contractor C provided the smallest duration discount. However, as the evaluators perceived contractor C to be highly likely to implement its commitments, contractor C had the highest expected utility value. Furthermore, contractor C's projected cost was also within the project's budget. The quality commitment of contractor A also had the highest expected utility value, which indicated that contractor A was the most likely to successfully implement the commitment.

Selection of the Optimal Contractor
The aforementioned expected utility analysis was also applied to evaluate the cost discount in the bid commitment. The bid commitments of the five candidate contractors were represented in terms of the overall weighted duration discount, d, cost discount, c, and quality, q. Their corresponding expected utility values, vd, vc, and vq, are presented in Table 15. The overall prospect utility value was calculated using the following equation: where v is prospect value; wd, wc, and wq are the duration, cost, and quality weights (values are 0.402, 0.302, and 0.296); vd, vc, and vq are expected utility values. As indicated in Table 15, the top three overall prospect values based on the posterior probability obtained from the BFPM were higher than 500. In other words, the BFPM can be used as a contractor selection model for committee members to define the minimum threshold when selecting contractors in the considered project of mass-rapid-transit station development. When six evaluators were considered in the proposed prediction model, contractor A attained the highest ranking, with the highest prospect utility value and received the highest evaluation. Moreover, contractor A also ranked first in quality, and its expected utility with respect to duration and cost was also above average. Therefore, contractor A was selected as the optimal applicant, followed by contractor E. Contractors B and C provided the lowest cost discount. As the probability of implementing a cost discount was very low, the prospect utility of cost commitment was very low. By contrast, contractor C's duration commitment was positively evaluated by the panel of experts. Therefore, contractor C ranked third with respect to the overall prospect utility. Although contractor B offered the highest duration discount, this contractor performed unfavorably in the possibility of implementing cost commitment. Thus, contractor B had the lowest expected utility. The expected utility of contractor B with respect to quality commitment was low. As the cost discount of contractor B was low, the implementation probability of cost commitment was also low. Therefore, contractor B had the lowest prospect utility, which indicates that the implementation of contractor B's bid commitment carries a high risk.
The results obtained from BFPM and from other contractor selection decision-making models were compared. If the risk in the (unsuccessful) implementation of bid commitment was not considered, the model providing the lowest tender or overall utility value [4] was adopted. Without this consideration of risk, contractor B won the bid and had the highest ranking (Table 14). When the successful implementation of bid commitment considered in the MCPM and BFPM was converted into risk probability, contractors A, C, and E were the top three contractors with the highest overall prospect values. The duration commitment of contractor C was recognized by the group of evaluators; thus, contractor C ranked third with respect to overall prospect utility. Although the discounts offered by contractors A and E did not match the highest discounts offered by contractors C and D, contractors C and D were less likely to implement their projects successfully. Therefore, both contractors had similar overall evaluation results. In BFPM, the provision of external information was incorporated into the risk probability. As contractor A had the highest prospect utility with respect to quality commitment, this contractor won the bid, followed by contractor E. In general, the contractor offering the highest overall utility may also carry a high risk of poor implementation, relative to the bidder's offers. This phenomenon can be quantitatively represented using the Bayesian posterior probability.
The use of FUT and the Bayesian posterior probability can increase the difference between the overall prospect values, which is in line with the idea of establishing minimum threshold selection criteria for committee members. This approach can be integrated in the selection criteria when releasing the tender documentation announcement. BFPM and MCPM can effectively filter out the options included in contractor selection criteria which carry a high cost risk, despite a high cost utility. Moreover, BFPM and MCPM do not cause malicious competition in the bidding process. In the BFPM, the posterior probability of external information in BT can increase the difference between the ratings of bidding contractors and truly reflect the requirements of establishing a minimum score threshold when public bidding involves procurement procedures. The results of BFPM method can avoid the lowest bidder being selected; in addition, the score gap of contractor selection can be increased from 4.12% (MCDM) to 22.87%, and the difference between the top three scores can be shortened from 71.91% (MCDM) to 30.04%.

Conclusions
In Taiwan, the most advantageous tender in governmental procurement is the selection of a general contractor based on score or ranking evaluated by committee. In fact, due to personal subjective preference, contractor selection of committee members may be different, causing cognitive difference between the results of the members' selection and the preliminary opinions provided by the working group. Thus, if the overall performance of bid contractors can be predicted before contractor selection, specifically with respect to duration, cost, and quality, the committee can better select the optimal general contractor for the construction project.
In this study, to solve the aforementioned contractor selection problem, we developed a new model, BFPM, which considers the three influence factors of duration, cost, and quality. Four main set evaluation methods were defined. The proposed method of integrating the probability by using Bayes' theorem was used to calculate the subjective utility of factors, which was assessed by using the fuzzy utility, before obtaining the prospect values for contractors multiplied their Bayesian probabilities by utilities. To obtain the overall estimate, fuzzy theory was also used to recalculate the objective weights and combine the subjective risk preference and objective utilities.
The obtained result provides a theoretical basis for using a method with wide practical applications for combining factor weights (obtained by using MPR and FPR methods) as the arithmetic mean of some sets of values. The considered methods use the posterior probability of BT to represent the probability of implementing bid commitment. This model aids committee members in their selection of the best contractor for public construction projects. The results of this study can avoid the lowest bidder being selected; in addition, the score gap of contractor selection can be increased, and the difference between the top three contractors' scores can be decreased. This method further verifies that the committee members establish the minimum threshold criteria of contractor selection. The results of this study not only ensure the duration and cost in public construction, but also promotes project quality. Moreover, our contributions aid sustainability in the operations and development of public infrastructure.
The proposed method combines the risk preference to calculate the factor weights and utilities after obtaining the estimates of a group of experts. The method also transforms subjective preference into objective weights and utilities. The calculated weights of the factors, the Bayesian probability, and the utility functions can serve as a reference example. This research focuses on the process of selecting optimal contractors, discussing the personal preferences of committee members, and analyzing the members' preference behaviors for contractors through a mathematical model (Bayes theorem et al.) Using the mathematical model, in addition to proposing an innovative decision-making system of contractor selection and an index weight-assessing system for sustainable development, this model will be widely applied and sustainably updated for other similar cases, such as railway station development, urban renewal or social housing buildings of contractor selection for public construction projects. The results of BFPM help to select the best contractor, can be applied to the life cycle construction, and can promote sustainable development. Funding: This study received no external funding.

Conflicts of Interest:
The authors declare no conflicts of interest.