Research on rating behaviors of bidding evaluation experts in construction projects based on novel integrated methods about reliability and decision entropy

Abstract

In recent years, with the rapid development of information technology and the demand for prevention and control of COVID-19, electronic bidding has become very popular, and remote construction project bid evaluation has been widely used. In practice, it is easy to detect and give early warning for propensity abnormal scores with large deviation values. Still, it is impossible to make timely judgments for non-pendency abnormal scores, so its reliability needs to be analyzed. This paper studied the rating behaviors of bidding evaluation experts in construction projects, and proposed novel integrated methods about reliability and decision entropy. Firstly the paper used the power method in numerical algebra to obtain the score closest to the real score of the project and then calculated the decision entropy of the individual expert, and the calculated entropy value corresponds to the reliability of the decision. Finally, the paper determined the warning threshold through instance analysis to monitor the reliability of evaluation expert scoring behaviors. The detailed methods and steps proposed in this paper are helpful in improving the management level of construction project bid evaluation.

Introduction

With the strong demand for electronization and informatization of bidding activities, enterprises have started to establish a unified public service platform for electronic transactions and to carry out the new practice of “contactless bid opening”. In this type of evaluation activity, the reviewing experts are the core. Under the premise of determining evaluation objectives, evaluation criteria, evaluation objects, and other evaluation elements, the evaluation results are completely determined by the evaluation behaviors of each expert (Wu and Tang, 1992). The fact shows that the remote bid evaluation of the construction project has played a beneficial role in the sharing of bid evaluation experts’ resources, preventing corruption and colluding in bidding and other aspects, thus promoting the bidding activities to be more fair and equitable. However, in practice, there are still some problems in remote bid evaluation. The professionalism and fairness of bid evaluation experts need to be measured, and it is difficult to ensure that the scores and suggestions given by bid evaluation experts are consistent with the strength of bidders (Li, 2002).

In practical work, due to various reasons such as aging knowledge structures, differences in values, and varying levels of familiarity with evaluation objects, some experts often experience irrational preferences and misplaced evaluation directions during the evaluation process, resulting in certain deviations in evaluation results. Therefore, reliability research of rating behaviors of bidding evaluation experts in construction projects is a very worthwhile research question (Tsung-Han and Tien-Chin, 2009).

If the deviation value of the abnormal score is large, it can be detected by using the mathematical statistics method. However, some bid evaluation experts will not score abnormally high or abnormally low but will control the score of all bidders within a reasonable range, their evaluation of the bidder’s ability is not scientific, and it may occur that the bidder with excellent ability scores lower than all their score, In this case, the scoring reliability of bid evaluation experts is poor, which can not truly reflect the actual ability of the bidder, and this problem is difficult to detect through mathematical statistics. Therefore, the reliability of the score that has not been detected as abnormal needs to be analyzed to warn those bid evaluation experts who have insufficient evaluation ability or have illegal behaviors (Liu et al., 1997; Wan, 2007; Liu and Lin, 2008).

Entropy measures the complexity of a time series and represents the degree of chaos in the system and can be used to define the uncertainty or reliability of information (Lei et al., 2021; Khaleie and Fasanghari, 2012). The bid evaluation of engineering projects is a group decision-making process, and the bid evaluation results are affected by the decision-making level of individual experts, who find it difficult to make perfect decisions and may even make mistakes. Therefore, the uncertainty of the evaluation conclusion of individual members, namely the decision entropy of individual experts, can be used to evaluate the reliability of experts (Xu, 2018).

Literature review

Some relevant literature has been analyzed as follows.

• (1) In terms of the risk of electronic bidding in construction projects, studied a real-time detection technology for the bid-winning and price-raising behaviors of online bidding projects, analyzed the risks generated in the process of electronic bid evaluation, and put forward countermeasures and suggestions. Gao (2017) analyzed the risks faced by electronic bidding from the aspects of environment, technology, and management, and put forward corresponding risk management measures. With the rapid development of science and technology, electronic bidding is increasingly applied to today’s bidding and trading activities. Traditional paper file archiving has been unable to meet the needs of electronic bidding and bidding, and electronic bidding and bidding archives have emerged as the times require. This article introduces the application of electronic archives in the two domestic electronic bidding and bidding platforms certified by Samsung, analyzes the characteristics and problems of electronic archives, and puts forward some thoughts on electronic archives (Zhao, 2022). Refining and chemical enterprises need to cooperate with suppliers through bidding in important work such as material and raw material procurement. As the market competition becomes more and more transparent, refining and chemical enterprises and related enterprises have also encountered many problems that are difficult to ignore in bidding management, leading to poor quality of bidding management and affecting the daily work of refining and chemical enterprises. This article analyzes several common problems in the bidding work of refining and chemical enterprises, According to the current market environment, optimization measures are put forward to improve the quality of bidding work in refining and chemical enterprises (Sun, 2023).

• (2) For the collusive behaviors of bidding participants, Huo and Zhang (2016) analyzed and studied the links and mechanisms of the collusive behaviors of bid evaluation experts in the bid evaluation process, established a variety of collusive game models, and pointed out that the main factors affecting the collusive behaviors of bid evaluation experts are the supervision, collusive interests, rent-seeking behaviors, punishment measures and reputation of the tenderer and the bidder. Chen and Fu (2020), based on constructing the conceptual model of influencing factors of collusive behaviors tendencies, used questionnaires to investigate relevant stakeholders in bidding and tendering and used a structural equation model and layer-by-layer regression method to analyze the questionnaire data, and tested the influence path of information symmetry, risk tendency, competitive pressure, moral concepts and other influencing factors on collusive behaviors tendencies. Gübilmez and Briain (2021) examined the bidding behaviors of institutional investors in initial public offering auctions using a hand-collected dataset of limit bids. Oo and Tang (2021) examined: (i) the level of bidding information feedback from public, private, and quasi-government client groups, and (ii) the sources, types, uses, and adequacy of bidding feedback information from these client groups from contractors’ perspective. Data was collected using an online questionnaire. Nosratabadi et al. (2022) proposed an efficient risk-based infrastructure management approach for a multi-energy microgrid to assess the effectiveness of demand-side management (DSM) through a stochastic strategy. The studied microgrid consists of photovoltaic and combined cooling, heat, and power as the energy generation units and load aggregator with the aim of DSM. Based on this structure, day-ahead operational scheduling is investigated for the microgrid in different scenarios in both summer and winter seasons. The risk consideration is also performed in the bidding procedure dealing with different scenarios and variable prices and probabilities.

• (3) Some information technology including big data can assist all parties in bidding and bid evaluation. For bid evaluation experts, big data technology can assist bid evaluation experts in bid evaluation. Wei (2018) pointed out that big data technology can be used to collect and analyze market material prices and other information to assist bid evaluation experts in judging whether the material cost price is too low and whether the bid price is lower than the cost price; It can also analyze the price of similar projects, scheme types, etc. so that the bid evaluation experts can grasp the evaluation criteria more reasonably and give accurate score to the bidders. Rublein et al. (2021) propose a two-round bidding approach of assigning tasks to edge cloud servers while taking into account various processing requirements and server constraints.

• (4) Regarding the reliability of expert scoring, Wu and Gao (2019) presented a reliability analysis method of bid evaluation experts based on generalization theory; in the generalization research stage, the evaluation indicators are taken as the fixed side, the evaluation experts are used as the random side, and the variance analysis is performed on the different evaluation index dimensions. Guillemard et al. (2024) aimed to assess intraobserver and interobserver reliability of a new lymphoscintigraphy quantitative and qualitative scoring system. Morison et al. (2024) presented the reliability method of the Wheelchair Error Scoring System among clinicians from multiple disciplines.

Big data technology also plays a great role in the supervision of bidding activities. Zhang (2016) believes that after the standardized collection of the basic data of bidding and tendering, big data technology can be used to analyze and judge the abnormal behaviors of the tenderer from the aspects of bid winning rate, project capital saving rate, ceiling price, bid winning price, etc., and analyze and judge the abnormal behaviors of the bidders from the aspects of the similarity of bidding documents, the correlation between bidders, IP address, etc The abnormal behaviors of bid evaluation experts are analyzed and judged in terms of abnormally high and abnormal low score.

Previous studies about construction project bid evaluation mainly focused on whether the lowest price is reasonable and scientific and various factors affecting the cost involved in bid evaluation. In terms of bid evaluation methods, the multi-attribute decision-making method was mostly used for research. According to the actual development and application of bidding, the following documents put forward the application of bid evaluation methods. The research is also increasingly diversified, especially the subjective evaluation of qualitative evaluation factors in the process of bid evaluation. The evaluation factors are determined by questionnaire interviews, case studies, data collection, and other forms, and a variety of certain multi-attribute decision-making models are used for relevant analysis and evaluation. It is rarely studied from the perspective of the reliability of the evaluation expert scores of construction projects. This paper analyzes the scoring reliability of construction project evaluation experts based on the method of decision entropy. The power method in numerical algebra is used to obtain the score closest to the real score of the project, and then the decision entropy of the individual expert is calculated. The reliability analysis results can more intuitively reflect the expert level.

This paper mainly includes the introduction, the reliability analysis method of bid evaluation expert scoring behaviors, the determination of early warning threshold and case analysis, research results, and some conclusions.

Reliability analysis method of bid evaluation expert scoring behaviors

Management of bid evaluation experts in the construction project bidding work is mainly based on the evaluation opinions of bid evaluation experts fed back by the tenderer and the dynamic evaluation score of bid evaluation experts from the administrative departments. These are only the evaluation from the external behaviors level of bid evaluation experts. The actual work level of bid evaluation experts has not been fully evaluated, and their scoring reliability and evaluation ability cannot be effectively evaluated. The bid evaluation experts have not been effectively supervised and managed. With the promotion and application of electronic bidding and big data technology, the scoring data of bid evaluation experts, as a huge repository of knowledge, can be more objectively and effectively evaluated by analyzing and mining the internal relationship of the scoring data and constructing the scoring reliability model of bid evaluation experts, to reflect the true level of bid evaluation experts to relevant administrative and regulatory departments, to better supervising and managing bid evaluation experts. Various evaluation activities and their management are long-term tasks, and the management of experts should also adopt a long-term, closed-loop approach. The reference criteria for forming the next evaluation expert group are also to some extent the basis for predicting the “level” of evaluation. The reliability of experts is ultimately reflected in the objectivity and accuracy of their evaluation opinions. The reliability of the bid evaluation experts’ score is the key to achieving the optimal selection of contractors. According to the relevant literature, there are 2 main methods for reliability analysis of bid evaluation experts’ scores.

• (1) Discreteness-based method

  The degree of dispersion refers to the measurement index of the difference in understanding between an expert and other experts when evaluating the project. Generally, the evaluation takes into account the opinions of most experts, so the statistical value of the difference between the scoring data of an expert and the final evaluation results of the project can be used as the basis for counter-assessment (Zhang et al., 2008). The higher the degree of dispersion is, the greater the difference between the expert and other experts in understanding the project evaluation, and the lower the accuracy of the evaluation is.

• (2) Method based on reliability coefficient

Reliability is an estimate of the degree of consistency of measurement, while expert reliability is the consistency of multiple experts’ scoring data for a project. The greater the reliability coefficient, the consistency between the evaluation experts is the higher. Like the dispersion degree, when the evaluation takes into account the opinions of most experts, the degree of consistency between the scoring data of an expert and the final evaluation results of the project can be used as the basis for evaluating the reliability of the evaluation expert scoring behaviors (Zhang et al., 2008). The definition method can be used to calculate the reliability coefficient, that is, the ratio of the true score variance in the measured score to the measured variance; It can also be expressed by the grade correlation coefficient, ranking the expert score and the final score of the project, and assigning the corresponding grade value, and then measuring the linear correlation between the expert score ranking and the final ranking of the project (Xie and Zhang, 2006; Zhang et al., 2008).

Method based on decision entropy

In combination with the above evaluation expert scoring behaviors reliability analysis method, the method used the dispersion or reliability coefficient is to use the existing data to measure the difference or consistency between an expert’s score and other experts. The smaller the difference or consistency, the higher the expert’s score level this time. However, there is a hypothesis in the calculation process of the dispersion and reliability coefficient that the average value of all experts’ scores is the real score of the project. The decision entropy method first uses the power method in numerical algebra to obtain the score closest to the real score of the project, and then calculates the decision entropy of the individual expert. The result is more accurate, and the calculated entropy value corresponds to the reliability of the decision. The reliability value range is [0, 100%], and the reliability analysis result can more intuitively reflect the expert level. Therefore, the paper chooses the method based on decision entropy to analyze the scoring reliability of construction project evaluation experts. The characteristics comparison table of three methods for evaluating the reliability of expert ratings is sown as Table 1.

Table 1 The characteristics comparison table of three methods for evaluating the reliability of expert ratings.

The process diagram of the decision entropy method is shown as Fig. 1.

Fig. 1

The process diagram of the decision entropy method.

Scoring reliability models and early warning threshold based on decision entropy

Assume $\{{S}_{1},{S}_{2},\cdots ,{S}_{a}\}$ expresses a decision group G formed by a bid evaluation experts form, $\{{D}_{1},{D}_{2},\cdots ,{D}_{n},\}$ represents the set of n bid documents to be evaluated, and x_ij represents the independent evaluation score given by expert S_i for the bid document D_j. The larger the x_ij, the better the bid evaluation expert S_i thinks the D_j of the bid document.

A vector formed by scoring results of Si evaluation expert for all bidding documents D_j is

$$xi=({x}_{{\rm{i1}}},{x}_{i2},\cdots ,{x}_{in})\in {E}^{n}$$

(1)

A matrix formed by scoring results of decision group G on all bidding documents D_j is

$$X=({x}_{1},{x}_{2},\cdots ,{x}_{n})={({x}_{ij})}_{a\times n}$$

(2)

The decision-making level of experts is affected by such factors as professional level, experience, knowledge reserve, emotion, and preference, and there will be some deviation when scoring compared with the ideal state, that is, there is no expert whose decision-making reliability reaches 100% in reality. the paper only considers the expert with the highest decision-making level, the most accurate rating, and the most fair, that is, the expert with the highest decision-making level as the ideal expert S*, and its scoring vector is

$${x}_{\ast }=({x}_{\ast 1},{x}_{\ast 2},\cdot \cdot \cdot ,{x}_{\ast n})\in {E}_{n}$$

(3)

Obviously, the lower the decision-making level of expert Si, the greater the difference between his conclusions and S*. Therefore, this “difference” can be used to measure the decision-making level of S_i, that is, the reliability of the evaluation expert scoring behaviors.

Score of solving ideal expert S_* based on characteristic root method (GEM) of group decision

The characteristic root method of group decision only requires the bid evaluation experts to score the bid documents directly, and then the score matrix is transposed and recorded as matrix F, and the eigenvector corresponding to the largest characteristic root of F is the optimal decision conclusion. The accuracy requirement is ε under the condition that, the power method in numerical algebra can be used to quickly calculate x, and the specific algorithm is as follows.

• (1) Let $k=0$, ${y}_{0}=\left(\frac{1}{n},\frac{1}{n},\cdot \cdot \cdot ,\frac{1}{n}\right)\in {E}_{n}$,${F}={x}^{T}\,{x}$,${y}_{1}={F}_{{y}_{0}}$, ${z}_{1}=\frac{{y}_{1}}{{\Vert {y}_{1}\Vert }_{2}}$;

• (2) Let $k=1,2,\cdot \cdot \cdot$; ${y}_{k+1}={F}_{{z}_{k}}$, ${z}_{k+1}=\frac{{y}_{k+1}}{{\Vert {y}_{k+1}\Vert }_{2}}$;

• (3) Use $|{z}_{k\to k+1}|$ to express the maximum absolute value representing the difference between the corresponding component, if $|{z}_{k\to k+1}| < \varepsilon$, that is all ${x}_{\ast }$, otherwise go to step ②.

Where x_ij represents the independent evaluation score given by expert S_i for the bid document D_j, n represents the total number of the bid document D_j, the values of other symbols can be obtained by calculating the above formula based on these two basic values, the meanings of these symbols can be found in section “Scoring reliability models and early warning threshold based on decision entropy”.

Unitization of scoring data

The score of bid evaluation experts is unitized, and the formula is

$${b}_{ij}=\frac{{x}_{ij}}{\sqrt{{x}_{i1}^{2}+{x}_{i2}^{2}+\cdot \cdot \cdot +{x}_{in}^{2}}}$$

(4)

Where $i=\ast ,1,2,\cdots ,a$.

Unit scoring vector of bid evaluation experts ${S}_{i}$${B}_{i}=({b}_{i1},{b}_{i2},\cdots ,{b}_{in})\in {E}^{n}$, The unitized results of the decision group G on all bid documents D_j forms a matrix $B=({b}_{1},{b}_{2},\cdots ,{b}_{n})={({b}_{ij})}_{a\times n}$.

Expert decision level vectors

Use N_i = (N_i1, N_i2,⋯, N_in) express Indicates the ranking of the bidding documents according to the experts’ score S_i (i = *, 1, 2⋯, a), record the bid document with the highest score given by experts as D_j*, position in a name list D_ij* = 1; record bid document with the lowest score D_j0, position in a name list N_ij0 = n, The remaining values N_in are the same; If there is the same score, the ranking shall refer to the ranking order of ideal experts S_*.

The expression of the decision level vector for the expert S_i is

$${E}_{i}=({e}_{i1},{e}_{i2},\cdot \cdot \cdot ,{e}_{in})$$

(5)

$${e}_{ij}=1-|{N}_{\ast j}-{N}_{ij}|-|{d}_{\ast j}-{d}_{ij}|$$

(6)

Where $i=\ast ,1,2,\cdot \cdot \cdot ,a$; $j=1,2,\cdots ,n$; ${e}_{ij}\in [1-n,1]$.

The decision level vector of the ideal expert is, and each component of the vector is 1, that is, it reaches the maximum value, indicating that its decision level is the highest. The smaller the value of each component of the expert’s decision level vector, the greater the difference between its decision judgment and the result given by the ideal expert, that is, the lower the reliability of the expert’s decision.

Expert S _i decision entropy

Through calculation, the sum of the generalized entropy of the components of the expert’s decision level vector can be obtained, which can be used as the individual decision entropy of the expert to describe the inaccuracy of the expert’s decision conclusion and measure its decision level. The specific algorithm is as follows.

$${H}_{i}=\mathop{\sum }\limits_{j=1}^{n}{h}_{ij}$$

(7)

$$\begin{array}{cc}{h}_{ij}=\left\{\begin{array}{l}-{e}_{ij}\,{\mathrm{ln}}\,{e}_{ij}\\ \frac{2}{e}-{e}_{ij}|{\mathrm{ln}}\,{e}_{ij}|\\ \frac{2}{e}\\ \frac{2}{e}-{e}_{ij}\sqrt{{\mathrm{ln}}^{2}(-{e}_{ij})+{\pi }^{2}}\end{array}\right. & \begin{array}{c}(\frac{1}{e}\le {e}_{ij}\le 1)\\ (0\, < \,{e}_{ij}\, < \,\frac{1}{e})\\ ({e}_{ij}=0)\\ (1-n\le {e}_{ij}\, <\, 0)\end{array}\end{array}$$

(8)

Where $i=\ast ,1,2,\cdots ,a$; $j=1,2,\cdots ,n$.

The less inaccurate the expert’s decision conclusion is, the more reliable the decision conclusion is and the lower the decision entropy is. This conclusion is consistent with the reality, that is, high-level bid evaluation experts have fewer decision-making mistakes, high reliability of decision-making conclusions, and low decision-making entropy (Qiu, 2011).

The expression of decision reliability R_i for expert S_i s is

$${R}_{i}=f({H}_{i})$$

(9)

The corresponding relationship between the decision-making reliability R_i of expert S_i s and some entropy values H_i is shown in Table 1. When H_i = 0, R_i = 100%; When H_i∈ [0, 0.34657n), R_i = 50%; when H_i∈ [0.0099498 n, 0.019799 n), R_i = 99%, and so on.

Decision reliability and entropy table is shown as Table 2.

Table 2 Decision reliability and decision entropy table.

The correction model of expert decision reliability

According to the provisions of the Regulations for the Implementation of the Bidding Law, after the completion of the bid evaluation, the bid evaluation committee shall recommend no more than three candidates for winning the bid and indicate the order. Therefore, as long as the top three bid evaluation expert’s can truly reflect the bidder’s ability, it will not affect the determination of the candidate for the winning bidder.

When the decision entropy method is used to measure the reliability of bid evaluation experts, when there are more than five bidders, even if the top three experts are consistent with the ideal expert’s bid evaluation results, but when the ranking of other bidders has a pair of adjacent data reversed, the decision entropy will be large due to the cumulative effect, and the measured reliability of bid evaluation experts will generally be low, and less than 60%. Considering that the Bid Evaluation Committee recommends no more than three candidates for winning the bid, only the sum of the decision entropy values of the top three bidders in the bid evaluation expert scoring behaviors is calculated, and the calculation formula is as follows.

$${H}_{i}^{\text{'}}=\mathop{\sum }\limits_{q=1}^{3}{h}_{iq}$$

(10)

Where H_iq represents the decision entropy of experts to the top three bidders.

$${H}_{i}^{{\prime\prime} }={H}_{i}-{H}_{i}^{{\prime} }$$

(11)

${H}_{i}^{{\prime\prime} }$ Represents the sum of decision entropy values of an expert ${S}_{i}$ to the top 3 bidders who are not rated

Build decision reliability correction model of expert ${S}_{i}$

$$\begin{array}{cc}R^{\prime} =\left\{\begin{array}{l}f({H}_{i})\\ f{({H}_{i}^{{\prime\prime} })}^{\frac{1}{n-3}}f({H}_{i}^{{\prime} })\end{array}\right. & \begin{array}{c}n\le 3\\ n > 3\end{array}\end{array}$$

(12)

Where the greater the value n, the smaller the impact ${H}_{i}^{{\prime\prime} }$ on reliability ${R}_{i}^{{\prime} }$.

Determination of early warning threshold and 2 case analyses

After obtaining the entropy value ${H}_{i}^{{\prime} }$, find the corresponding reliability ${R}_{i}^{{\prime} }$ in combination with Table 1. According to most principles in the determination method of early warning threshold, the early warning threshold value of reliability is set to be 80%, that is, when ${R}_{i}^{{\prime} }\ge 80 \%$, the evaluation expert’s scoring reliability is considered to be high, and the scoring can accurately reflect the true ability of the bidder, without early warning; When ${R}_{i}^{{\prime} }$ is less than 80%, it is considered that the scoring reliability of the bid evaluation experts is low and the evaluation level of the experts is not high, and early warning should be given to the bidding administrative supervision department.

Case 1

A construction bidding project of railway engineering has 10 units bidding, and the bid evaluation committee is composed of 7 experts. Experts are selected from the expert database. In practical work, some experts may even give unreasonable evaluations. Various evaluation activities and their management are long-term tasks, and the management of experts should also adopt a long-term, “closed-loop” management approach. The reference standards for forming the next evaluation expert group are also to some extent the basis for predicting the “level” of evaluation (Liu et al., 1997; Wan, 2007; Liu and Lin, 2008).

The scoring table for the project bidding proposal in case 1 is shown as Table 3.

Table 3 The scoring table for the project bidding proposal in case 1.

The detailed information of the experts is shown in Table 4.

Table 4 The detailed information table of the experts in case 1.

According to Table 2 and Formula (2), the first step is to calculate the matrix.

$$F={x}^{T}x=\left[\begin{array}{cccccccccc}68649.37 & 67497.14 & 66872.24 & 66537.53 & 66980.23 & 67246.80 & 68891.11 & 66756.58 & 68154.81 & 66899.41\\ 67497.14 & 66371.43 & 65750.17 & 65413.11 & 65850.43 & 66112.34 & 67733.88 & 65630.80 & 67011.60 & 65769.64\\ 66872.24 & 65750.17 & 65148.07 & 64812.26 & 65253.62 & 65511.30 & 67107.37 & 65043.59 & 66391.37 & 65173.54\\ 66537.53 & 65413.11 & 64821.26 & 64507.55 & 64934.16 & 65190.75 & 66772.86 & 64717.21 & 66058.27 & 64857.16\\ 66980.23 & 65850.43 & 65253.62 & 64934.16 & 65365.66 & 65623.29 & 67216.25 & 65146.17 & 66498.45 & 65286.42\\ 67246.80 & 66112.34 & 65511.30 & 65190.75 & 65623.29 & 65882.92 & 67484.20 & 65403.43 & 66762.99 & 65544.42\\ 68891.11 & 67733.88 & 67107.37 & 66772.86 & 67216.25 & 67484.20 & 69134.32 & 66992.12 & 68394.44 & 67135.89\\ 66756.58 & 65630.80 & 65034.59 & 64717.21 & 65146.17 & 65403.43 & 6992.12 & 64928.50 & 66276.26 & 65068.24\\ 68154.81 & 67011.60 & 66391.37 & 66058.27 & 66498.45 & 66762.99 & 68394.44 & 66276.26 & 67665.58 & 66417.55\\ 66899.41 & 65769.64 & 65173.54 & 64857.16 & 65286.42 & 65544.42 & 67135.89 & 65068.24 & 66417.55 & 65208.79\end{array}\right]$$

Where x_ij represents the independent evaluation score given by expert S_i for the bid document D_j, n represents the total number of the bid document D_j, the values of other symbols can be obtained by calculating the above formula based on these two basic values, the meanings of these symbols can be found in section “Scoring reliability models and early warning threshold based on decision entropy” and “Score of solving ideal expert S_* based on characteristic root method (GEM) of group decision”.

According to Formula 3, The second step is the calculation of ${x}_{\ast }$.

In terms of accuracy requirements ε = under 0.0001, the results of each iteration using the power method are shown in Table 5.

Table 5 Results of each iteration.

From the last column of Table 4,

x_* = (0.3218 0.3164 0.3135 0.3120 0.3140 0.3153 0.3230 0.3130 0.3195 0.3136)

ε = 0.0001 is the ideal expert scoring behaviors vector x_*, and the bidding document score from high to low are D₇å D₁å D₉å D₂å D₆å D₅å D₁₀å D₃å D₈å D_4.

The third step is the calculation of ${N}_{i}$.

$${N}_{i}=\left[\begin{array}{ccccccc}2 & 2 & 2 & 2 & 2 & 4 & 2\\ 5 & 10 & 4 & 3 & 4 & 10 & 4\\ 8 & 9 & 6 & 5 & 5 & 9 & 6\\ 10 & 6 & 10 & 10 & 10 & 8 & 10\\ 6 & 7 & 7 & 7 & 7 & 3 & 7\\ 4 & 3 & 5 & 6 & 6 & 2 & 5\\ 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 9 & 8 & 9 & 9 & 9 & 7 & 9\\ 3 & 5 & 3 & 4 & 3 & 5 & 3\\ 7 & 4 & 8 & 8 & 8 & 6 & 8\end{array}\right]$$

Since the bid evaluation experts have the same score for different bidding documents, their ranking is ${N}_{i}$ obtained by referring to the ranking order of ideal experts ${S}_{\ast }$ (According to 2.3).

The fourth step is to calculate the unit score matrix B^T according to Formula (4).

$${B}^{T}=\left[\begin{array}{ccccccc}0.3232 & 0.3205 & 0.3241 & 0.3203 & 0.3217 & 0.3178 & 0.3252\\ 0.3158 & 0.3107 & 0.3216 & 0.3188 & 0.3192 & 0.3081 & 0.3209\\ 0.3120 & 0.3124 & 0.3128 & 0.3149 & 0.3180 & 0.3122 & 0.3122\\ 0.3115 & 0.3159 & 0.3084 & 0.3123 & 0.3094 & 0.3165 & 0.3094\\ 0.3143 & 0.3146 & 0.3104 & 0.3142 & 0.3146 & 0.3185 & 0.3114\\ 0.3159 & 0.3169 & 0.3137 & 0.3142 & 0.3146 & 0.3185 & 0.3130\\ 0.3248 & 0.3234 & 0.3258 & 0.3215 & 0.3217 & 0.3185 & 0.3253\\ 0.3118 & 0.3146 & 0.3104 & 0.3142 & 0.3114 & 0.3169 & 0.3114\\ 0.3192 & 0.3162 & 0.3241 & 0.3175 & 0.3201 & 0.3177 & 0.3219\\ 0.3153 & 0.3169 & 0.3104 & 0.3142 & 0.3114 & 0.3177 & 0.3114\end{array}\right]$$

Step 5, calculates the unitized scoring matrix E_i according to formula (5) and formula (6).

$${E}_{i}=\left[\begin{array}{ccccccc}0.9986 & 0.9987 & 0.9977 & 0.9985 & 0.9999 & -1.0040 & 0.9967\\ -0.0006 & -5.0057 & 0.9949 & -0.0024 & 0.9972 & -5.0083 & 0.9955\\ 0.9985 & -0.0011 & -1.0007 & -2.0013 & -2.0045 & -0.0013 & -1.0014\\ 0.9996 & -3.0039 & 0.9965 & 0.9997 & 0.9975 & -1.0046 & 0.9974\\ 0.9997 & -0.0006 & -0.0036 & -0.0002 & -0.0006 & -2.0044 & -0.0027\\ -0.0006 & -1.0016 & 0.9984 & -0.0010 & -0.0007 & -2.0032 & 0.9977\\ 0.9981 & 0.9995 & 0.9972 & 0.9985 & 0.9988 & 0.9955 & 0.9978\\ 0.9989 & -0.0016 & 0.9974 & 0.9987 & 0.9984 & -1.0039 & 0.9984\\ 0.9996 & -1.0033 & 0.9954 & -0.0021 & 0.9994 & -1.0018 & 0.9976\\ 0.09998 & -2.0032 & -0.0033 & -0.0006 & -0.0023 & -0.0040 & -0.0023\end{array}\right]$$

The sixth step is to calculate the decision entropy H_i, ${H}_{i}^{{\prime} }$, ${H}_{i}^{{\prime\prime} }$ of bid evaluation experts. Firstly, calculate ${h}_{ij}$ according to formula (8).

$${h}_{ij}=\left[\begin{array}{ccccccc}0.0014 & 0.0013 & 0.0023 & 0.0015 & 0.0001 & 3.8899 & 0.0033\\ 0.7408 & 18.4079 & 0.0051 & 0.7519 & 0.0028 & 18.4183 & 0.0045\\ 0.0015 & 0.7439 & 3.8795 & 7.1747 & 7.1854 & 0.7455 & 3.8816\\ 0.0004 & 10.7346 & 0.0035 & 0.0003 & 0.0025 & 3.8917 & 0.0026\\ 0.0003 & 0.7403 & 0.7592 & 0.7376 & 0.7407 & 7.1853 & 0.7537\\ 0.7408 & 3.8824 & 0.0016 & 0.7286 & 0.7411 & 7.1810 & 0.0023\\ 0.0019 & 0.0005 & 0.0028 & 0.0015 & 0.0012 & 0.0045 & 0.0022\\ 0.0011 & 0.7472 & 0.0026 & 0.0013 & 0.0016 & 3.8895 & 0.0016\\ 0.0004 & 3.8877 & 0.0046 & 0.7500 & 0.0006 & 3.8832 & 0.0024\\ 0.0002 & 7.1812 & 0.7571 & 0.7405 & 0.7514 & 0.7612 & 0.7514\end{array}\right]$$

According to formula (7), formula (10), and formula (11), respectively calculate H_i, ${H}_{i}^{{\prime} }$, ${H}_{i}^{{\prime\prime} }$.

${H}_{i}$ = (1.4887 46.3269 5.4185 10.8879 9.4271 49.8501 5.4055)

${H}_{i}^{{\prime} }$ = (0.0036 3.8842 0.0097 0.7549 0.0019 14.3707 0.0079)

${H}_{i}^{{\prime\prime} }$ = (1.4851 42.4427 5.4087 10.1330 9.4252 35.4794 5.3976)

Step 7: calculate the reliability ${R}_{i}^{{\prime} }$ of bid evaluation experts.

In combination with Table 1, calculate the reliability ${R}_{i}^{{\prime} }$ of evaluation experts’ scores according to Formula (12), as shown in Table 6.

Table 6 reliability values ${R}_{i}^{{\prime} }$ of bid evaluation experts.

Case 2

The total land area of the X Square construction project is 207790 m², with a total building area of 791160.97 m². The above-ground building area is 627174.50 m² (residential area 503893.67 m², kindergarten 3600 m², commercial and other above ground capacity area 119680.83 m²), and the underground building area is 63986.47 m² (underground garage 154986.47 m², supermarket 9000 m²). The building density is 24.31%, the plot ratio is 3.02, and the greening rate is 35%. X Square construction project of railway engineering has 10 units bidding, and the bid evaluation committee is composed of 7 experts. See Table 2 for each.

The actual evaluation data for the fourth phase of the bidding stage of the X Square commercial district construction project is shown in Table 7.

Table 7 the scoring table for the project bidding proposal in case 2.

The detailed information of the experts is shown in Table 8.

Table 8 the detailed information table of the experts in case 2.

Set precision requirements ε = 0.0001

• (1) According to Table 5 and Formula (2) to calculate the unit rating vector of the expert

$${\rm{F}}={x}^{T}x=\left[\begin{array}{cccccccccc}65316.16 & 65229.41 & 66972.63 & 66529{\rm{.88}} & 65055.13 & 65254.19 & 65633{\rm{.96}} & 65884{\rm{.87}} & 67257{\rm{.22}} & 64947{\rm{.15}}\\ 65229.41 & 65145.23 & 66884.71 & 66442{\rm{.43}} & 64970.31 & 65166{\rm{.70}} & 65547.93 & 65795.96 & 67169.70 & 64864.18\\ 66972.63 & 66884.71 & 68687.83 & 68232.05 & 66707.24 & 66916.94 & 67302{\rm{.53}} & 67576.02 & 68979.54 & 66595.80\\ 66529.88 & 66442.43 & 68232.05 & 67780.93 & 66266.28 & 66474.41 & 66857.42 & 67128.10 & 68521.96 & 66155.30\\ 65055.13 & 64970.31 & 66707.24 & 66266.28 & 64796.65 & 64994.00 & 65372.59 & 65623.82 & 66991.06 & 64689.66\\ 65254.19 & 65166.70 & 66916.94 & 66474.41 & 64994.00 & 65198{\rm{.96}} & 65572{\rm{.85}} & 65835{\rm{.48}} & 67200{\rm{.92}} & 64884.62\\ 65633.96 & 65547.93 & 67302{\rm{.53}} & 66857.42 & 65372.59 & 65572.85 & 65954.83 & 66208.63 & 67588.79 & 65264.72\\ 65884.87 & 65795.96 & 67576.02 & 67128.10 & 65623.82 & 65835.48 & 66208.63 & 66490.68 & 67862.10 & 65510.84\\ 67257.22 & 67169.70 & 68979.54 & 68521{\rm{.96}} & 66991.06 & 67200.92 & 67588.79 & 67862.10 & 69273.01 & 66879.86\\ 64947.15 & 64864.18 & 66595.80 & 66155.30 & 64689{\rm{.66}} & 64884{\rm{.62}} & 65264{\rm{.72}} & 65510.84 & 66879.86 & 64584.67\end{array}\right]$$

• (1) k = 0

  ${{u}}_{0}^{{\rm{T}}}=[\begin{array}{cccccccccc}\frac{1}{10} & \frac{1}{10} & \frac{1}{10} & \frac{1}{10} & \frac{1}{10} & \frac{1}{10} & \frac{1}{10} & \frac{1}{10} & \frac{1}{10} & \frac{1}{10}\end{array}]$

• (2) k = 1

  ${u}_{1}^{T}={(A{u}_{0})}^{T}=\left[\begin{array}{cccccccccc}65808.06 & 65721.66 & 67485.53 & 67038.87 & 65546.67 & 65749.91 & 66130.42 & 66391.65 & 67772.41 & 65437.68\end{array}\right]$ ${z}_{1}^{T}=\frac{{u}_{1}}{||{u}_{1}|{|}_{2}}=\left[\begin{array}{cccccccccc}0.3138 & 0.3134 & 0.3218 & 0.3197 & 0.3126 & 0.3135 & 0.3154 & 0.3166 & 0.3232 & 0.3121\end{array}\right]$

• (3) k = 2

$${u}_{2}^{T}={(A{u}_{1})}^{T}={10}^{10}\times \left[\begin{array}{cccccccccc}4.3642 & 4.3585 & 4.4755 & 4.4459 & 4.3469 & 4.3604 & 4.3856 & 4.4030 & 4.4945 & 4.3397\end{array}\right]$$

$|{z}_{2\to 1}|=0.0027\times {10}^{-4}$, $|{z}_{2\to 1}| < \varepsilon$, ${z}_{2}^{T}$ is the rating vector of the expert ${T}_{\ast }$${v}^{\ast }=\left[\begin{array}{cccccccccc}0.3138 & 0.3134 & 0.3218 & 0.3197 & 0.3126 & 0.3135 & 0.3154 & 0.3166 & 0.3232 & 0.3121\end{array}\right]$_.

The degree of superiority and inferiority of the bidding documents from high to low is:

D₉ > D₃ > D₄ > D₈ > D₇ > D₁ > D₆ > D₂ > D₅ > D₁₀.

$${z}_{2}^{T}=\frac{{u}_{2}}{{\Vert {u}_{2}\Vert }_{2}}=\left[\begin{array}{cccccccccc}0.3138 & 0.3134 & 0.3218 & 0.3197 & 0.3126 & 0.3135 & 0.3154 & 0.3166 & 0.3232 & 0.3121\end{array}\right]$$

Where x_ij represents the independent evaluation score given by expert S_i for the bid document D_j, n represents the total number of the bid document D_j, the values of other symbols can be obtained by calculating the above formula based on these two basic values, the meanings of these symbols can be found in section “Scoring reliability models and early warning threshold based on decision entropy” and “Score of solving ideal expert S_* based on characteristic root method (GEM) of group decision”.

Following the same method as case 1, the _rating reliability of the evaluation expert T_i can be obtained as shown in Table 9.

Table 9 the rating reliability of the evaluation expert T_i.

Results

In case 1, it can be seen from Table 4 that the scoring reliability of bid evaluation experts S₁, S₃, S₅, and S₇ is higher than 80%, and their scoring reliability is acceptable without warning; The scoring reliability of bid evaluation experts S₂, S₄ and S₆ is lower than 80%, and their scoring reliability is too low, which requires early warning to the bidding administrative supervision department.

In case 2, the reliability of the scores given by the evaluation experts T₁, T₂, T₃, T₅, and T₇ is all above 80%, and their reliability is considered normal; The reliability of the ratings of evaluation experts T₄ and T₆ is both below 80%, indicating a low level of reliability and abnormality, which requires special attention.

In the decision entropy method, the reliability of its research results and discoveries is ensured through the participation of evaluation experts in multiple evaluation activities, including Case 1 and Case 2 shown in this paper,

The results of this article are compared with other methods as follows. The method based on dispersion or reliability coefficient uses existing data to measure the difference or consistency between the rating of a certain expert and other experts. The smaller the difference or the greater the consistency is, the higher the rating level of the expert this time is. However, there is a hypothesis in the calculation process of dispersion and reliability coefficient, and the average score of all experts is the true score of the project; The decision entropy method first uses the power method in numerical algebra to obtain the score closest to the true score of the project, and then calculates the decision entropy of individual experts, resulting in more accurate results.

The paper adopted the decision entropy analysis method to analyze the decision-making methodology, pricing characteristics, and evaluation quality of bidding and tendering. In addition, the paper also aimed to enhance the awareness and understanding of maximizing management efficiency among all parties. By comprehensively considering the bidding characteristics of bidders and the decision-making behaviors of evaluation experts, a corresponding model was constructed, which can provide a valuable reference for the strategic behaviors of bidding parties and therefore has certain theoretical significance.

The paper provided an in-depth exploration of the normative practice of bidding quotation decision-making and evaluation behaviors, which is of great significance for the future development of each construction enterprise and even the fate of the entire industry. Against the backdrop of intensified competition in the current construction market, conducting such research has profound practical significance.

Conclusions

This paper studied the evaluation expert scoring behaviors’ reliability analysis method, used the decision entropy method to build a scoring reliability model, and determined the early warning threshold to monitor the evaluation expert scoring behaviors’ reliability. First of all, for the abnormal score caused by non-principle reasons, which may be caused by the mistakes of bid evaluation experts or insufficient evaluation ability, will be included in the scoring reliability analysis as the basic data for analyzing the evaluation ability of bid evaluation experts. Then, for the score that is not abnormal after detection, the decision entropy method is used to analyze its reliability. When the reliability R_i is less than 80%, the evaluation expert’s scoring reliability is considered to be low, that is, the expert’s evaluation level is not high, and an early warning should be sent to the bidding administrative supervision department. The method and detailed steps proposed in this article have been validated in the X Square construction project. The method proposed in this article does not have any special requirements, so it can also be used in other similar construction projects. In practical applications, the detailed algorithm steps provided in this article, such as the power method in numerical algebra, information entropy calculation, and decision opinion generation, can all be implemented using programs, which can to some extent solve the problem of computational efficiency when dealing with large datasets.