contractors are operating at very high risks. Even worse, some final bid prices are so low that barely covers the total cost. Under such circumstances, it is very likely to see problems such as low-quality work, schedule delay or even contractor bankruptcy.
To avoid the over-competition and super-low final bid price, the government agencies are empowered to decide if the lowest bidder is able to win the bid if the bid offer is less than 80% of the government’s base price. To judge whether the final bid price is reasonable or not is a very difficult task and often times the decision is made solely based on personal experience, judgement or preference. It is the purpose of this research to develop a final bid price prediction model to assist the final tendering decision making process for the government agencies. Utilizing the available information such as budget amount, bond price, base price, number of bidders and project duration, final bid price prediction models will be developed using Batch Least Squares Estimation (LSE) method.
In order to do so, information from a total of 1240 public construction projects from 2006 to 2010 is collected to develop and validate the proposed model. For the developed model, five variables (owner’s base price, budget amount, bond price, number of bidders and project duration) are set as the independent variables and the final bid price is set as the dependent variable. Initially, Pearson Correlation Analysis is conducted to examine the relationships between individual independent variable and the dependent variable. The results have shown that owner’s base price, bond price and budget amount are most related to dependent variable (final bid price) and therefore, are used to be the model inputs in this research. Among the 1240 project collected, two-thirds of them are used to train and validate the model and the remaining one-third of them are used to test the model.
The analysis results show that, comparing to the multiple regression results, the Batch Least Squares Estimation model render better outputs when the equation is set to higher power (i.e. forth power is better than cubic or square). Projects are grouped into five groups by their total cost, and individual prediction models are developed for each group. The best results obtained from the Whole-Batch Least Squares Estimation models provide valuable information to the government agencies when they are making the tendering decision. If the lowest bid price offered by the contractor deviates greatly from the model prediction, cautious should be taken since the bid price might not be reasonable. In addition, the contractors can adopt the same model to make final bid price prediction.
This serves as valuable reference for the contractors when they are coming up with their final bid price in the bid preparation stage.
2 Literature Review 2.1 Tender awarding methods
According to the Taiwan Public Procurement Laws, there are four methods for tender awarding:
lowest bid, most reasonable bid, most advantageous bid, and multiple winning bids. For lowest bid tender awarding, the bidder with lowest bid will be awarded the contract under the circumstance that their bid proposal must be lower than the authorized base price. For contracts without authorized base price, the lowest yet reasonable bid will win the contract if the bid is lower than the budget amount. If the above two methods are not applicable, contracts will be awarded to the most advantageous bidder judged by a panel of reviewers based on not only the price, but also qualification, past experience in similar projects, company finance, professional capability and so forth. For multiple winning bids, the contract can be awarded to more than one bidder for different combination of procurement items or procurement quantities. However, the principles of lowest bid or most advantageous bid should also apply in the multiple winning bids. This procurement law regulates procurement for goods, services and construction projects. In practice, most construction projects in Taiwan adopt the lowest bid contracting strategy and this research will only focus on this tender awarding method.
Except for turnkey projects, most construction projects in Taiwan are delivered in the fashion of design-bid-build. Unless there are dramatic changes in the scopes after the design is completed, these projects are awarded to the contractor using the lowest bid contracting strategy. Nevertheless, the lowest bid method often provokes fierce and sometimes, destructive competition among the bidders.
Some opportunistic bidders might render unreasonably low bid just to win the project and problems such as claims, disputes, delays and cost overruns are more likely to occur in the project execution phase (Williamson, 1975). Wang (2004) has pointed out in his research that the final bid prices for public construction projects in Taiwan tend to fall on the low side and when the final bid prices are too low, it is questionable whether the contractors are able to maintain the quality of the final works.
Although the government agencies have the right to not awarding the contracts if they determine the lowest bid price is unreasonable, it is a very difficult decision to make without proper reference. It is the goal of this research to develop a final bid price prediction model to serve as valuable reference for the contract awarding decision-making process.
2.2 Bid price prediction models
Under the Taiwan Procurement Laws, if the lowest bid price is lower than 80% of the authorized base price, the government agencies are allowed to award the contract to the second lowest bidder if they have enough reasoning to believe the lowest bid is not reasonable. Nevertheless, very few contracts are awarded to the second lowest bidder because there is no guarantee that the second lowest bidder will accomplish the work on time, under the budget and fulfill the quality requirements. Furthermore, if the contract was awarded to the second lowest bidder instead of the lowest bidder, there might be suspicions that the government agencies profiteered for favoring the second lowest bidder. As a result, if the lowest bid is unreasonable low, the contract will still be awarded to that bidder with an extra discrepancy bond deposited in the amount of the difference between 80% of the base price and lowest bid price. In light of this, it is important to have final bid price prediction models that are able to provide valuable information for reference when evaluating if a winning bid is reasonable or not.
Hsu (2009) used multiple regression analysis and extended Kalman Filter to develop final bid price and authorized base price prediction models for roadway projects. Data from 400 projects are collected for analysis. Factors such as budget amount, project duration and bid bond are used to predict the final bid price and authorized base price. The research indicates that prediction results obtained are better for authorized base price models comparing to final bid price prediction models.
Hung (2007) collected data from a total of 187 projects to develop models for predicting the indirect cost of a project. Three factors, project type, project duration and project scale (represented by direct cost) are set as the input for the prediction model. Three techniques, statistical regression, artificial neural network (ANN) and fuzzy ANN are adopted for model development. The research has indicated that models developed using fuzzy ANN yield the best prediction results. The results provide valuable reference for the contractors when they are estimating the indirect cost.
Chu (2007) collected data from 2251 roadway construction projects between 2001 and 2006 to study the relationship between authorized base price and final bid price. Stepwise regression and cluster analysis are conducted for the analysis. The results have shown that when the number of bidder increases, the difference between authorized base price and final bid price also increases. In addition, when the construction cost price index gets higher, the difference between the authorized base price and final bid price gets smaller. The government agencies are able to use the prediction models when they are setting the base price or evaluating whether the final bid price is too low. Chen et al. (2005) studied 82 roadway construction projects to predict the final bid price. Statistical regression techniques are applied to build the model and factors such as budget amount, project duration and bid bond are set as the independent variables. The results show that the prediction model yields the best results when project duration and bid bon are used to predict the final bid price.
Based on the literature above, several researches have been conducted to develop prediction models for project bid price. Traditional statistical regression methods and artificial intelligence methods (ANN, fuzzy ANN) are adopted to develop the prediction models. For this research, five factors, budget amount, bid bond, authorized base price, project duration and number of bidders, will be set as the independent variables for the final bid price prediction models. In the mean time, Batch Least Squares Estimation method will be adopted for the model development.
2.3 Whole-Batch Least Squares Estimation method
First introduced by Carl Friedrich Gauss to describe the behavior (orbits) of the celestial bodies, the least squares method adjusts the parameters of a model function to best fit a dataset (Bretscher, 1995).
For example, a set of n observed data points with independent variable xi and dependent variable yi
can be denoted as:
(xi, yi), i = 1, 2, 3, ……, n (1)
The objective of least squares method is to find the parameter values for the function y = f (x) that
“best fits” the observed data. The optimum (or best-fit) is achieved by minimizing the squared residuals, as shown in Eq. 2.
min. ( y
i− f x ( )
i)
2i=1
∑
n (2)The equation can be expressed by polynomial equation with a set of parameter to be determined. If time sequence is not of concern, the parameters can be directly computed from the “batch” of data that is loaded at one time for the equation. The upside is that there is enough data to obtain the parameters more accurately but the downside is that the computation time is long and there is need for larger memory space for the computer (Robert, 2006). If the observed values cannot be obtained at one time, recursive least squares estimation can be applied. Existing data is used to estimate the parameters and these parameters are then adjusted with new data. The downside is that the results are not precise enough at the early stage with recursive least squares estimation method (Grant, 1987).
Least squares estimation has been used in many different fields for function parameter estimation.
Plett (2010) demonstrates how total least squares method gives better results than traditional methods when computing the total battery cell capacity estimate. Wang et al. (2011) propose a generalized asymmetric least squares regression method to estimate value-at risk and expected shortfall. Their results with S&P 500 stock index inputs clearly show that the method is superior to other existing benchmark methods. Kanamori et al. (2009) propose a least squares approach to estimating the ratio of two probability density functions. The proposed method is able to provide a closed-form solution and is computationally highly efficient. In addition, their numerical experiments have shown that the accuracy of the proposed method is comparable to the best existing method. Brus and Gruijter (2011) introduce design-based Generalized Least Squares (GLS) estimation of status and trend of soil property at selected time points from monitoring data collected in repeated soil surveys with partial overlap. The above literature has pointed out that Least Squares Estimation can be applied in many different fields and the results obtained are promising. Therefore, this research intends to adopt Least Squares Estimation to predict the final bid price for the public construction projects.
3 Model Development
The basic principles behind the Least Squares Estimation are to find a set of function parameters that minimize the squared residuals between observed values and function estimated values. The final bid price prediction model development is briefly described below.
For this research, five factors are adopted as the independent variable after extensive literature review is conducted. These five factors are budget amount (M), bid bond (N), authorized base price (O), number of bidders (P) and project duration (Q). For linear estimation, the final bid price (Z) for project i can be expressed as: