BIDDING DECISION MAKING FOR CONSTRUCTION COMPANY USING A MULTI-CRITERIA PROSPECT MODEL

Two critical decisions faced by bidders in competitive bidding include, firstly, whether or not to submit a bid, and secondly (if the answer to the first is ‘yes’) what markup value should be used on the submitted bid. In the construction industry, government agencies and private sector clients typically adopt competitive bidding to determine contract awards. Contractors also apply the same approach to bidding decisions. There are many variables that affect contractor decisions regarding whether to bid and the markup scale, which complicate the bidding decision process. This study proposes a Multi-Criteria Prospect Model for Bidding Decision (BD-MCPM) to assist contractors to make decisions on bid/no bid and markup scale. Key factors of influence that impact bidding decisions were identified first. Second, Fuzzy Preference Relations (FPR) was employed to assess factor weights and determine bid/no bid. Finally, if a decision to bid is given, then the Multi-Criteria Prospect Model (MCPM), which links Fuzzy Preference Relations (FPR) and Cumulative Prospect Theory (CPT), is deployed to determine the markup scale that best conforms to primary decision maker (PDM) preferences. The applicability of this model was demonstrated in a real case study.


Introduction
In the construction industry contractors typically earn construction contracts through either direct negotiation or competitive bidding. Government agencies and private sector clients most often employ competitive bidding, which commonly use the lowest bid price as the main award criterion. Usually the bid price includes cost of construction and a markup, the scale of which is typically determined using a percentage of construction costs. Size of markup impacts upon the profit, which serves as the primary motivator for a contractor to win and execute a contract (Dikmen et al. 2007). Research in the area of competitive bidding strategy models has been conducted since the 1950s (Friedman 1956). Numerous models have been developed, some of which were designed specifically for the construction industry. Despite the number of competitive bidding strategy models that have been developed, few of these are used in practice, largely as they do not address the practical needs of construction contractors (Hegazy and Moselhi 1995;Shash 1995). Therefore, there is a perceived need for models designed in line with actual construction contractor practices. In the bid process, once a determination is made to bid, the next step is to select an appropriate markup (Egemen and Mohamed 2008). A successful contractor is the one that selects the most optimal bid markup that secures both the contract and contract profitability (Shash and Abdul-Hadi 1992). Bid markup decisions currently follow no accepted standards or formal procedures, but rather consider contractor experience, intuition, and personal preferences, which are not conducive elements for building an effective approach for achieving the optimal bid markup (Chua and Li 2000).
Cumulative prospect theory was proposed by Tversky and Kahneman (1992). Different from the classical theory, CPT adopted a concave-shaped utility function (UF) for gains and convex for losses and an inverse S-shaped probability weighting function (PWF) to describe individual preferences for choosing between risky prospects. Wakker and Deneffe (1996) proposed a tradeoff (TO) method to improve probability distortions and misconceptions in utility elicitation. Many studies (Wu and Gonzalez 1996;Gonzalez and Wu 1999) have worked to elicit the PWF for particular subjects. Abdellaoui (2000) used TO method concepts to propose a parameter-free method to elicited subjects' UF and PWF. Bleichrodt and Pinto (2000) also leveraged the concept to propose a parameter-free method somewhat different from Abdellaoui's study, which they applied successfully to medical decision making. Determining the relative weight of influencing factors is important in multi-criteria decision making (MCDM). For uncertain events, the decision maker will find it difficult to form a judgment by relying on exact numerical values. FPR is a useful tool to express decision maker's uncertain preference information and define the relative weight of influencing factors. Significant attention has been given to fuzzy preference relations in previously studies (Orlovsky 1978;Nurmi 1981;Tanino 1984;Kacprzyk 1986;Chiclana et al. 1998Chiclana et al. , 2001Chiclana et al. , 2003Fan et al. 2002;Da 2002, 2005;Herrera-Viedma et al. 2004). Wang and Chang (2007) adopted FPR to forecast the probability of successful knowledge management.
The usual practice is to make bid decisions based on 'intuition', which can be described as a derivation of 'gut feelings', experience and guesswork (Ahmad 1990 Contractors currently make bidding decisions using several relevant models. Early mark-up scale estimation models (e.g., Friedman 1956;Gates 1967;Carr 1982) employed probability theory to predict the probability of winning a particular contract. However, as the bidding decision is a complex decision-making process affected by numerous factors, probability theory is unable to describe interactions between factors.
Researchers have recently introduced bidding decision support systems based on artificial intelligence (AI), which permit consideration of identified factors of importance. Such systems include the expert system (ES) (Ahmad and Minkarah 1988;Tavakoli and Utomo 1989), case-based reasoning (CBR) (Chua et al. 2001), neural network (NN) (Li 1996;Moselhi et al. 1993;Hegazy and Moselhi 1994;Dias and Weerasinghe 1996;Wanous et al. 2003), analytical hierarchy process (AHP) (Seydel and Olson 1990;Cagno et al. 2001), and fuzzy set theory (Eldukair 1990;Fayek 1998;Lai et al. 2002;Lin and Chen 2004). The ES is one of rule-based systems. The process of bid decisions are highly unstructured, uncertainty, and subjectivity. It's too complicated to creating a set of clear rules that would be suitable for all/most cases. CBR requires a reasonably large set of cases data from which to draw knowledge to avoid generating inaccurate results. NN, also called artificial neural network (ANN), is similar to the CBR, with an important exception that the inference process is concealed from the decision maker. For such reasons, NN-derived conclusions are sometimes not particularly convincing to decision makers. AHP is a decision-making approach that structures multiple-choice criteria into a hierarchy and assesses relative importance of each.
Unfortunately, AHP employs a complicated process to obtain consistent assessment results, which makes it unwieldy in practice.
The complexity and hard-to-define nature of competitive situations necessitates that most bid decisions rely heavily on decision maker intuition, experience and guesswork (Ahmad 1990). Fuzzy set theory provides a useful tool to handle decisions in which phenomena are imprecise and vague. Eldukair (1990) integrated fuzzy set theory with a multi-criteria model to select bidding cases. Subsequently, Fayek (1998) and Lai et al. (2002) used fuzzy set theory to choose optimal mark-up scales. Lin and Chen (2004) proposed an approach using fuzzy set theory to obtain a linguistics suggestion result for a bid/no-bid selection.
Fuzzy preference relations (FPRs), which integrate fuzzy logic and AHP concepts, greatly improve on AHP in terms of relative weight evaluation. In BD-MCPM, the FPR is used to determine the relative weights of influencing factors, and the CPT is used to evaluate the PDM's preference. The BD-MCPM handles factors marked by relatively higher levels of vagueness to make complicated bidding decisions and determine PDM risk preference. Results conform to actual bid decisions generated based on decision maker intuition, experience and guesswork. Therefore, BD-MCPM can assist decision makers to identify projects with the greatest profit potential and set an optimal mark-up scale.

Fuzzy preference relations
Most decision processes are based on preference relations (PR), the most common representation of information in decision making. In PR, an expert assigns a value to each pair of alternatives that reflects the degree of preference for the first alternative over the second. Many important decision models have been developed using mainly two preference relation types: (1) Multiplicative Preference Relations (MPR) and (2) Fuzzy Preference Relations (FPR).
Most decision processes are based on preference relations, the most common representation of information in decision making. An expert assigns a value to each pair of alternatives that reflects degree of preference B as an alternative over others. Many important decision models have been developed using mainly: (1) multiplicative preference relations and (2) fuzzy preference relations (Herrera-Viedma et al. 2004).
A multiplicative preference relation on a set of alternatives X is represented by matrix A, with A usually assumed a multiplicative reciprocal: The a ij indicate the preference ratio of alternative x i to x j . Saaty (1980Saaty ( , 1994 suggested measuring a ij using a ratio scale 1-9. When a ij = 1 indicates indifference between x i and x j , and a ij = 9 indicates that x i is absolutely preferred to x j , then Fuzzy preference relation B on a set of alternatives X is a fuzzy set on the product set X×X, characterized by membership function µ B : X×X→[0,1]. Therefore: where µ B is a membership function and b ij is the preference ratio of alternative x i over x j . A b ij at 0.5 denotes that x i and x j are indifferent, and a b ij at 1 represents that x i is preferred absolutely to x j . The method (Herrera-Viedma et al. 2004) to transformation multiplicative preference relations A to fuzzy preference relations B and obtain relative weights presents the following: (1) Get (n-1) values {a 12 , a 23 , …, a (n-1)n } of multiplicative preference relations A; (2) Diagonal elements of A are values at 1.0, using the equation (5) to calculate the remaining elements in the upper right part of the diagonal: ( 1) ( 1) Elements in the lower left part of the diagonal in were calculated using the equation shown below: (3) Let Z = max[A], then transfer multiplicative preference relations A to a consistently MPR matrix C with a normal to interval [1/9, 9] with equation (7): (4) Apply equation (8) to transform the consistent MPR matrix C to FPR matrix B:

Cumulative prospect theory
Consider a prospect (mutual fund) X with outcomes Cumulative prospect theory predicts that people will choose prospects based on the value (Tversky and Kahneman 1992): where 0 λ > is a loss-aversion parameter and π are decision weights calculated based on "cumulative" probabilities associated with the outcomes. In particular, prospect theory assumes a probability weighting function CPT the utility function ( ) v x is unchanged from the original PT (Tversky and Kahneman 1992;Tversky and Fox 1995;Gonzalez and Wu 1999), which is concave for gains and convex for losses, with the loss function assumed to be steeper than the gain function ( 1 β > ): The decision weights employed in CPT are given by Tversky and Kahneman (1992): where probability weighting functions w − and w + are defined for probabilities associated with losses and gains, respectively, which may be experimentally estimated using the following formulae (Tversky and Kahneman 1992;Camerer and Ho 1994;Wu and Gonzalez 1996): For only gain conditions, equations (9) will transform to: 3. Constructing a multi-criteria prospect model for bidding decisions

Multi-criteria prospect model for bidding decision
This study adopted BD-MCPM, which combined FPR and CPT, to modeling the construction company's bidding decision processes using the three phases shown in Fig. 1.

Phase I -preparation
The bidding decision process generates two decisions: whether to submit or not submit a bid (bid/no bid) and, if so, the scale of the markup component of the bid (markup) (Egemen and Mohamed 2008). Many factors affect decision making in each phase. Phase I should first identify the key factors that influence a bidding decision and, based on such factors, collect and organize relevant project data/information.   Cook (1985); (2) Skitmore (1985); (3) Marsh (1989); (4) Cooke (1992); (5) Shash (1993); (6) Odusote and Fellows (1992); (7) Wanous et al. (2000); (8) Chua and Li (2000); (9) Han and Diekmann (2001); (10) Lewis (2003) 3.2.1. Identify the key factors of influence in a bid decision Many studies designed to identify the factors that influence bidding decisions have been conducted in recent years. Some have adopted a contractor perspective. Others have focused on conditions limited to a particular, localized situation. Still others have taken a multinational perspective. All have worked to identify key factors of influence at work on local contractor bid decisions. The purpose of Phase I in the BD-MCPM was to identify, respectively, the key factors influencing bid/no bid and markup decisions. The identification process was a two step process, which first reduced the total potential number of factors by identifying and choosing only those referenced consistently in the literature in order to identify a shortlist of 'pre-adapted' factors. The second step incorporated these pre-adapted factors into a questionnaire, which was send to local contractors who were asked to assess the importance of each factor on a scale from 1 to 9 (1:very unimportant, 9:very important). Each factor was then assigned an importance score based on an average of submitted scores. Table 1 shows 44 factors identified in the literature as affecting bid/no bid decision making (Cook 1985;Skitmore 1985;Marsh 1989;Cooke 1992;Odusote and Fellows 1992;Shash 1993;Wanous et al. 2000;Chua and Li 2000;Han and Diekmann 2001;Lewis 2003). Sixteen of these factors were prioritized as they were mentioned in five or more of the referenced articles. Ten of these prioritized factors received average scores of importance equal to or greater than 5, and were ranked from highest to lowest.
Similarly, Table 2 shows the eight factors identified in the literature as affecting markup decisions (Odusote and Fellows 1992;Dozzi et al. 1996;Li 1996;Dulaimi and Shah 2002). A shortlist of those that were mentioned in two or more articles was then made, and those from the shortlist with average earned scores of importance equal or greater than 5 were ranked.

Case collection
A case study to test the ability of the BD-MCPM model to solve the above problem was conducted to illustrate the effectiveness of the approach in practice. The background of research participants were considered to be homogeneous in the sense that they were all qualified professionals in construction field with previous knowledge of bidding strategies and bidding procedures. Table 3 presents a summary of data collected on three actual projects. Prior project markup scale Common markup 3-6% Common markup 3-6% Common markup 3-6% The best case 20% gain The best case 20% gain The best case 20% gain The worst case 15% loss The worst case 15% loss The worst case 15% loss 3.3. Phase II -deciding to bid or not to bid The goal of Phase II was to make a decision whether or not to bid on a particular project. The 10 key factors that affect the bid/no bid decision were identified in Section 3.2.1. By assessing the relative weights and risk scores for these factors, a bid/no bid score may be obtained, which can then be used to make the decision whether to bid or not.
3.3.1. Determining the relative weight of influencing factors for bid/no bid decisions The seven steps employed to determine the relative weight of identified factors of influence are described as follows: Step 1: Define linguistic variables. This study used 9 linguistic terms {AM: Absolutely more important, VM: Very strongly more important, SM: Strongly more important, WM: Weakly more important, EQ: Equally important, WL: Weakly less important, SL: Strongly less important, VL: Very strongly less important, AL: Absolutely less Important } associated with real number {5, 4, 3, 2, 1, 1/2, 1/3, 1/4, 1/5} to compare corresponding neighboring factors.
Step 2: Obtain questionnaire input. Ten factors of influence [BF i (i=1,2,…,10)] were considered in making the bid / no bid decision. Via a questionnaire survey or interviews, the th k evaluator assessed the relative intensity of importance of the two adjoining factors BF i and BF j to obtain 9 grades of importance Step 3: Construct the MPR matrix. To construct the th k evaluator's MPR matrix k A , we first translated the linguistic terms of questionnaire results into real numbers k ij a to fill proper diagonal elements, using Eqs (5) and (6) to calculate the remaining elements of MPR matrix.
For example, from Step 5: Aggregate the FPR matrix for all evaluators. The opinions of different evaluators were aggregated to obtain an aggregated weight for each factor of influence. k ij b was employed to denote the fuzzy preference relationship value of the th k evaluator to assess factors i and j. This study used an average value method to integrate the judgment values of m evaluators and obtain the averaged FPR matrix B . The average function is shown below: Step 6: Normalize the aggregated FPR matrix. Using R to indicate the normalized aggregate FPR matrix, the value of element r ij can be obtained using the function shown below: Step 7: Obtain relative weights. Given that i WB denotes the priority weight of influencing factor i, the priority weight of each factor may be obtained using the following function:  (19) obtains a relative weight for influencing factor BF 1 of 0.123. Following the same process, the relative weights of influencing factors in Case 1 as assessed using three evaluators were obtained as WB i ={0.123, 0.074, 0.049, 0.082, 0.124, 0.114, 0.115, 0.093, 0.125, 0.101}.

Assessing the risk score for factors of influence in bid/no bid decision making
Risk score RS i represents the degree of risk in the factor of influence BF i , which has been subjectively established by PDM using predetermined scores {0 -No risk, 25 -Low risk, 50 -Moderate risk, 75 -High risk, 100 -Prohibitive risk}. For Case 1, the value of risk associated with influencing factors using PDM is illustrated in Table 5.

Deciding to or not to submit a bid
A total bid/no bid score may then be calculated by summing degrees of significance: If 50 B V ≤ , then a "bid" decision is recommended. Applying equation (20) to bid/no bid scores for cases 1 through 3 returned, respectively, total scores of 43.8, 52.9 and 45.3. As such, the contractor should bid on Case 1 and Case 3 and proceed to Phase III (the markup phase) for both.

Phase III -deciding appropriate markup
Once a decision to bid has been made, the next step is to determine appropriate markup sizes for bidding projects. Firstly, the PDM assign a set of markup scales. The evaluator then determines the relative weight of each factor of influence (see also Section 3.2.1) for each special markup scale. The resulting assess the probability of winning project for specify markup scale. The determining process is illustrated below.

Assign specific scale of markup
In construction projects, the scale of a markup is determined based on relevant contractor policies and project type. In general, markups tend to represent 3% ~ 7% of a project's total estimated cost, although in certain cases, markups may be in the 10% ~ 15% range or higher. This study adopted 5 frequently used markup scales {M 1 =3%,

Determine relative weight of influencing factors for special markup scale
The eight key factors previously identified as affecting markup scale decision making (MF i ) are listed in Table 2. These also represent factors of influence on outcome implementation. Assigning weights to each factor is done in the same manner as that described in Section 3.3.1. The only difference was the associated factors of influence used. The relative weight of each influencing factor MF i , assessed using 3 evaluators, were obtained and represented as WM i ={0. 078, 0.078, 0.064, 0.123, 0.191, 0.118, 0.150, 0.198}. 3.4.3. Forecast probability of winning project using markup scale As bids typically involve multiple potential contractors, assessing the probability of bid success over competitors at a particular markup level is critical. Of course, markup scale may be expected to correlate inversely to probability of bid success. FPR was used here to forecast project bid success in the same manner as in Section 3.3.1.
For each case and defined markup scale, evaluators used linguistic terms to judge subjectively the relative importance of each factor in winning a bid and in losing a bid MF i .
Using questionnaire results, a 2×2 pair-wise comparison MPR matrix could then be constructed with two outcomes ("win" and "lose") for each factor of influence. The pair-wise comparison MPR matrix B k for each influencing factor was then constructed (see Table 6 below). Applying the same process described from step 4 to 7 in Section 3.3.1., derived the average rating PR i , which described the potential for winning in light of the identified factors of influence MF i . For example, in Case 1, at a markup of 3%, win probability ratings for relevant factors of influence MF i may be obtained using PR i ={0. 81, 0.76, 0.79, 0.69, 0.78, 0.76, 0.83, 0.76}. Finally, for a specify markup scale, the forecast probability of winning PM may be obtained using the following function: where WM i and PR i denote the relative weights and the probability ratings of winning for identified markup scale factors of influence ( ) 1 2 8 i MF , i , , , ∈ L . Using the example of Case 1 at a 3% markup and the value for WM i obtained in Section 3.4.2., we may apply equation (21) to obtain a win probability forecast PM = 78%. In the same manner, the win probability forecast at 4%, 5%, 7% and 10% markups were 71%, 63%, 43% and 25%, respectively, for Case 1. In Case 3, win probability forecasts were 77%, 68%, 60%, 46% and 28% for defined markups in the 3~10% range.
As defined in the model construct, probability of winning a project is kept and probability of losing a project is ignored, the latter yields a prospect value equal to 0.

Elicit the PDM utility function for the markup scale
This study adopted the TO method proposed by Wakker and Deneffe (1996) to elicit the PDM utility function for the markup scale. This paper will not describe the mechanisms by which such was accomplished, as the method has been described previously in the literature (Bleichrodt and Pinto 2000;Abdellaoui 2000;Abdellaoui et al. 2005). The elicited result for the PDM utility function is shown in Fig. 2. ' w p . Fig. 3 shows the elicited PWF of the PDM in this study.
The calculated CPT value for each markup scale in Case 1 and Case 3 were listed in Table 7.

Comparison and decision making
Selecting the highest markup scale CPT value (Table 7) determined the markup scale in each case (i.e., 5% for Case 1 and 7% for Case 3). Estimated profit and bid price for Cases 1 and 3 were calculated and are shown in Table 8. Under circumstances in which contractors may only choose one case on which to bid, other consideration factors may be brought into play (e.g., duration, funding requirements, etc.).

Discussions
BD-MCPM was used successfully to help PDM determine which case(s) should be bid and the optimal markup. Knowing competitor markup scales prior to tender submission would be helpful in modifying the markup recommendations generated by BD-MCPM and allow for adjustments critical to winning the bid (markup adjustment downward) or increasing profit (markup adjustment upward). In practice, however, it is difficult to elicit a competitor's UF and PWF. Therefore, an effective methodology with which to infer such represents a valuable direction for future research.

Conclusions
This study developed a Multi-Criteria Prospect Model for Bidding Decision (BD-MCPM) to help contractors determine whether to submit a bid and, when the answer is in the affirmative, set an optimal markup scale. Research contributions include: 1. Identification of ten and eight key influencing factors used by contractors in Vietnam to make decisions, respectively, on bid/no bid and mark-up scales using literature review and questionnaire survey techniques.
2. Introduction of a new Multi-Criteria Prospect Model for Bidding Decision (BD-MCPM), which combines fuzzy preference relations (FPR), cumulative prospect theory (CPT), and Multi-Criteria Prospect Model (MCPM). The BD-MCPM is a systematic bidding model designed to help construction companies make strategic bid / no bid decisions and to determine the optimal markup scale for each project bid.
3. FPR using only a small number of expert input variables provides consistency in fuzzy preference relations that simplifies the process of evaluating relative factor weights to deciding bid/no bid phase and forecast probability project win for specific mark-up size. Moreover, applying FPR to evaluation and forecasting can connote the characteristic of evaluator's "experience" and "guesswork". 4. CPT evaluates the primary decision maker's risk prospects in terms of utility functions and probability weighting functions. CPT calculates preference values for assigned mark-up scales and probability of a project win based on prior forecasts. It further selects the mark-up scale delivering the optimal preference value so that the decision maker can make an optimal decision that takes into account the PDM's intuition.
5. The study validated the BD-MCPM using actual bidding projects obtained from construction companies in Vietnam and successfully helping the PDM to select cases on which to bid and to set optimal markup scale.