risk-based approach to unbalanced bidding in construction projects

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Risk-based approach to unbalancedbidding in construction projectsAbbas Afshar a & Helia Amiri aa Department of Civil Engineering and Center of Excellence forFundamental Studies on Structural Mechanics , Iran University ofScience and Technology , Tehran, IranPublished online: 22 Feb 2010.

To cite this article: Abbas Afshar & Helia Amiri (2010) Risk-based approach to unbalancedbidding in construction projects, Engineering Optimization, 42:4, 369-385, DOI:10.1080/03052150903220964

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Engineering OptimizationVol. 42, No. 4, April 2010, 369–385

Risk-based approach to unbalanced bidding inconstruction projects

Abbas Afshar* and Helia Amiri

Department of Civil Engineering and Center of Excellence for Fundamental Studies on StructuralMechanics, Iran University of Science and Technology, Tehran, Iran

(Received 7 February 2009; final version received 15 July 2009 )

Uncertainties in quantities of work items and vagueness in unit price boundaries have mainly beendisregarded in unbalanced bidding models. This article presents a fuzzy linear programming (FLP) modelwhich treats the quantities of works and unit price limitations as fuzzy numbers. Fuzzy constraints areused to define upper and lower bounds on the unit prices proposed by the contractor. The uncertaintiesin quantities of works are addressed through fuzzy coefficients with asymmetric fuzziness ratio in theobjective function. Therefore, the problem lends itself to an FLP model with fuzzy coefficients in theobjective function and fuzzy constraints. To account for these uncertainties, two different approaches areproposed and tested. Performance of the model is demonstrated using three hypothetical case examplesand the results are discussed and compared with those of a deterministic model.

Keywords: unbalanced bidding; uncertainties; fuzzy constraints; fuzzy linear programming

1. Introduction

When quantities of works are measurable for tendering, unit price contracts (UPC) are oftenemployed. To develop a UPC, work items and their estimated quantities for project executionare needed. These initially announced and/or estimated quantities help the contractor to comeup with beneficial unit prices for bidding. The contractor determines his total bid consideringthe proposed unit price for different items, the owner’s estimated quantities, and a fixed markupto cover overhead costs, profit and contingencies. The bid markup estimation for constructionprojects is challenging work in itself. Identifying the various factors affecting the level of risk,opportunity and competition in a project tendering, a well developed decision support system mayhelp the contractor for bid markup estimation (Dikmen et al. 2007).

Under some circumstances, the contractor may benefit from uneven markup distribution amongthe project activities. However, unit prices must be chosen in such a way that the summation of itemprices does not exceed the total bid price. Therefore, some activities may receive relatively higherand lower markups to retrieve (Cattell et al. 2007). In a typical unbalanced bidding, contractors

*Corresponding author. Email: [email protected]

ISSN 0305-215X print/ISSN 1029-0273 online© 2010 Taylor & FrancisDOI: 10.1080/03052150903220964http://www.informaworld.com

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370 A. Afshar and H. Amiri

tender relatively higher unit prices for work items scheduled for early stages of completion. Onthe other hand, items scheduled for later stages receive proportionately lower bidding unit prices.This policy may improve the company’s cash inflow and help the manager to invest the earlyreceived money on later stages of project implementation.

There are some other situations where unbalanced bidding could be advantageous. For instance,the contractor’s own estimation on the work items may differ from those outlined in proposal orannounced by the owner. In this case, a well planned and managed unbalanced bidding may bebeneficial by overpricing items assumed to be underestimated, while under-pricing those itemsassumed to be overestimated. Once again, proposing uneven distribution of markup among projectitems may be justified. Although there can be substantial risk due to the uncertainties involved,this strategy can help the contractor to submit a lower bid or improve the profitability of his tender.

The early version of unbalanced bidding was proposed by Gates (1959, 1967). Later, the basicconcept and its application and modelling scheme were modified and/or improved (Stark 1968,1972, 1974, Diekmann et al. 1982, Cattell 1984, 1987, Tong and Lu 1992). In a most recent work,Cattell et al. (2008) proposed an unbalanced bidding model incorporating all three kinds of itemprice loading. They illustrated that maximization of bottom-line profit may equally be replacedby maximization of the project’s top-line revenue.

Almost all previous models have been developed in deterministic environments. In a compre-hensive survey on unbalanced bidding approaches, Cattell and his colleagues criticize the previousresearches on the unbalanced bidding approaches (Cattell et al. 2008). According to the article,there are several statements which imply the lack of studies considering risks and uncertaintiesin previous unbalanced bidding models. Gates (1967) comments that there is considerable riskfor a contractor that his predictions regarding the anticipated variations may not be fulfilled.Notwithstanding this caveat, he does not propose any basis by which to measure or address thisrisk. Cattell (1984) shares the flaw of the other models by being dependent on the imposition ofupper and lower limits on unit price for each item. As with the other researchers, Cattell (1984)does not propose any definite mathematical modelling scheme for replacement of these arbitrar-ily chosen limits. Cattell (1987) identifies a single, and more comprehensive, formulation whichcomprises numerous variables that are of an uncertain nature. He argues that the various degreesof uncertainty in those variables contribute to an uncertainty in the formulated profit. Althoughhe notes that one may use simple mathematics to quantify a project’s overall risk, the proposedmodels do not incorporate any assessment of the contractor’s risk. It must be mentioned thatCattell (1987) uses modern portfolio theory to manage the decisions regarding the combination ofthe risks and returns generated by item pricing. It is clear that they have not explicitly accountedfor the uncertainties in the structure of the bidding model. Therefore the only model which openlytook uncertainty into account was the model proposed by Diekmann et al. (1982). However, theirmodel considers only uncertainties in quantities of work using a probabilistic approach. In a con-struction industry the probabilistic approach may fail due to lack of data and unique features ofconstruction projects. In a most recent article Cattell et al. (2008) propose a simplified unbalancedbidding model, where they have explicitly stated that the limits on the unit prices may be regardedas more fuzzy than fixed values. Based on the findings, they propose that future work may focuson the development of a complementary risk model, the purpose of which is the quantification ofthe combined risks that are generated by way of different item price combinations. This model,when used in combination with the model proposed here, can then be employed to maximizeprofit and minimize risk. They suggest that these two models should be combined using modernportfolio theory (MPT), together with indifference mapping and expected utility theory.

In reality, however, uncertainties are involved in many modelling parameters and/or variables.Generally, uncertainties arise from unquantifiable, incomplete, unobtainable information, and/orpartial ignorance. Uncertainty generally implies that one is unsure of the particular value a variableand/or parameter will assume. A variable’s value can be uncertain, both, if it is single-valued,

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Engineering Optimization 371

deterministic, and constant but its value is not perfectly known, or if its value is constantlyfluctuating with a random pattern (Lund 1991). Most of the effective parameters and/or variablesin an unbalanced bidding model are potentially subject to uncertainty. Some of them may beclassified as inherent random variables, while others might be treated as single-valued-unknownparameters. Interactions between these uncertain parameters make the precise modelling schemevery difficult, if not practically impossible. For example, uncertainties in quantities of workscan affect duration of the activities which may in turn change the project schedule and the actualimplementation unit costs. Even the activity duration itself cannot be defined as a certain parameter.Furthermore, the interest rate which is used to calculate the present value of profit is generallysubject to uncertainty.

Quantities of some work items such as soil and rock excavation cannot be estimated accuratelybefore the work is completely done. Although this item may be realized as an uncertain parameterand dealt with as a random number, it is actually a single-valued deterministic parameter whose realvalue is not perfectly known when the decision is being made. It seems that none of the researchershas considered this uncertainty except Diekmann et al. (1982) who improved Stark’s originaldeterministic model by adding a probabilistic formulation to account for risk of uncertainties inquantities of works. In addition to the probabilistic analysis, fuzzy sets, analytic hierarchy process(AHP), sensitivity analysis, and ‘robust’ analysis have been proposed which integrate uncertaintyinto decision analysis (Lund 2008). While these methods have been proved to be useful in someissues, all of them may be inferior to a properly formulated and applied probabilistic risk analysis,provided that reliable probability density functions are available.

Maximization of the present value of profit may result in unit prices far from those acceptableto the owner. Therefore, the owner may reject the proposed unbalanced unit prices if they exceedthe rational and reasonable ranges provided by the experts and/or authorities. The more theydeviate from the base price, the less they would be acceptable to the owner. Owing to the factthat the degree of acceptability is a vague statement by nature, definition and consideration ofthe reasonable unit price boundaries in the model may limit the contractor’s risk. On the otherhand, the structure of the model dictates most of the items to be priced at either the upper or lowerbounds which the contractor may impose as the constraints. Therefore, the optimum solutionto the model would highly depend on these values. As these bounds have no specific basis andthe degree of acceptability is a vague statement by itself, inclusion of the uncertainties in themodelling structure would beneficially provide a much more complete and real picture of theproblem than the conventional approaches in which these uncertainties are often ignored.

Every construction project has its own unique features; hence, time and cost for a given optionmay significantly vary from one project to another. To integrate existing uncertainties into deci-sion analysis, one must employ the most appropriate technique which best fits the nature of theprevailing uncertainties. Although the probabilistic risk analysis is reported to be superior to mostof the common risk analysis techniques (Lund 2008), its application is limited to the cases wherehard-to-get reliable probability density functions (PDF) are at hand. In fact, construction of suchPDFs for quantities of works needs adequate and precise data from similar projects implementedin quite similar environments and working conditions. However, owing to the uniqueness of eachconstruction project and unique features of every certain contract, collecting such information isvery difficult, if not impossible. In such cases, expert estimations on the range of cost of options(and/or activities) may be the most useful and dependable information. Therefore, a fuzzy-basedapproach may help to account for the uncertainties involved in unbalanced bidding approaches.

In 1970, a decision making process was presented by Bellman and Zadeh in a fuzzy environment.During recent decades various versions of fuzzy linear programming have been presented toaccount for different uncertainties and/or imprecision in goals and constraints (Tanaka and Asai1984, Ramik and Rimanek 1985, Chanas and Kuchta 1994, Sou-Sen Leu et al. 1999). Karimi et al.(2007) formulated and solved a fuzzy linear programming (FLP) model of earthwork allocations

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assuming fuzzy unit cost coefficients and borrow pits/disposal sites capacities while minimizingtotal earth-moving cost as an objective function. They recommended extending the problem forsimultaneous consideration of cost coefficients and right hand side of the constraints as fuzzynumbers. A very simple FLP model for unbalanced bidding was proposed by Afshar and Amiri(2008) in which quantities of works were assumed as triangular fuzzy numbers with symmetricfuzziness ratio on both sides.

This article presents a fuzzy linear programming (FLP) model to account for the most determi-nant uncertainties in an unbalanced bidding model. The model defines the upper and lower boundson the unit prices as fuzzy constraints, whereas uncertainties in quantities of works are consideredthrough triangular fuzzy coefficients with asymmetric fuzziness ratio in the objective function.The duration of activities, and so the project’s duration, are assumed to remain unchanged. There-fore, the problem lends itself to an FLP model with fuzzy coefficients in the objective functionand fuzzy constraints. To account for these uncertainties, two different approaches are proposedand tested.

2. Deterministic unbalanced bidding model, general structure

The most recent formulation for an unbalanced bidding is proposed as (Cattell et al. 2008):

PV =I∑

i=1

N∑n=1

(1

1 + ri

)n

[λniQi((1 + γnif )(1 − Rn)Xi − C ′′ni) + R′

n]

Where C ′′ni = Ci + C ′

ni

(1)

Subject to:n∑

i=1

Q′iXi = B. (2)

In which i = item number, n = month number, I = total number of work items, N = total durationof the project in months, ri = discount rate appropriate to item i, λni = proportion of Q′

i scheduledfor month n, Xi = bill price per unit of item i (decision variable), Ci = unit cost of the ith item,C′

ni = actual increase in the unit cost of item i in month n, C′′ni = actual inflated unit cost of

item i in month n. Q′i = bill quantity of item i (measured by client), Qi = bill quantity of item i

(measured by contractor), Rn = proportion retained in month n, R′n = the amount (if any) released

from the retention fund in month n including any interest earned, γ ni = adjustment for inflation,B = desired total bid f = proportion of the price that is contractually subject to escalation andPV = present value of profit.

Equation (2) guarantees unit prices are chosen in such a way that their sum, when multipliedby the owner’s quantity estimations, equals the total bid.

In reality, unit prices are not unbounded and may only vary within pre-specified limits and/orbounds. The risk of proposal rejection by client will increase as proposed unit prices violate thebounds. To account for the risk of proposal rejection, these uncertainties and bounds should beembedded into the modelling structure. Therefore, to be protected against quantity misestimatesor to hide pricing policies, and make sure unit prices are relevant to the direct costs, equation (3)is included to force unit prices fall within specific bounds (Stark 1968):

Li ≤ Xi ≤ Ui (3)

where Li and Ui are lower and upper bounds for work item i, respectively.

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In practice, unit prices of certain activities cannot exceed those of some relevant activities inthe same group. As an example, unit price of open excavation cannot logically exceed that ofunderground excavation. Equation (4) accounts for this rational constraint in unit prices (Stark1968):

Xa − Xb ≥ 0 (4)

where Xa and Xb = unit price for two similar items in the same group at different conditions.In reality, only an estimate of work quantities is available and their real values remain unknown

until the job is fully implemented. In other words, the quantities of works are single-valuedunknown parameters. On the other hand, upper and lower bounds on the unit prices may not bedefinitely fixed. Therefore, the uncertainties in the work quantities and bounds on the unit pricesmay be dealt with by employing fuzzy set theories in defining the objective function and fuzzyconstraints, respectively.

3. Unbalanced bidding model with fuzzy quantities and fuzzy constraints

With uncertain quantities of works expressed by fuzzy numbers, the objective function (Equation(1)) may be presented as:

Max Z =I∑

i=1

N∑n=1

(1

1 + ri

)n

[λniQi((1 + γnif )(1 − Rn)Xi − C ′′ni) + R′

n]. (5)

Replacing the deterministic upper and lower bounds on unit prices with fuzzy numbers, Equations(2)–(4) may be modified as:

n∑i=1

Q′iXi = B (6)

Li ≤ Xi (7)

Xi ≤ Ui (8)

Xa − Xb ≥ 0 (9)

Q = (Q, Q, m, n)L−L (10)

where the coefficient Qi defines a fuzzy number of the type (L −L).As defined by Dubois and Prade (1988), a fuzzy number Q is of L-R type if there exist shape

functions L (for left) and R (for right) and four parameters (Q, Q) ∈ R2, m, n and the membership

function of Q is:

μQ(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

L

(Q − x

m

)for x ≤ Q

1 for Q ≤ x ≤ Q

R

(x − Q

n

)for x ≤ Q

(11)

The fuzzy number is then denoted by Q = (Q, Q, m, n)L−R . If Q is a ‘trapezoidal fuzzy number’,

L(x) = R(x) = max(0, 1 − x) is implied. As the left and right hand side functions are the same,it is called ‘fuzzy number of the type (L −L)’ (Figure 1).

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Figure 1. Typical membership function of a trapezoidal fuzzy number (Afshar et al. 2008).

Therefore, m and n are non-negative real numbers and parameters Q and Q refer to the left and

right borders of the fuzzy number Q corresponding to the maximum reliability level (i.e. α = 1),respectively. Assuming m = kQ, coefficient k is defined as the left fuzziness ratio.

The problem defined by Equations (5)–(10) can be associated with a set of problems, whichdepend on a parameter α ∈ (0, 1] as follows (Chanas and Kuchta 1994):

Max Z =I∑

i=1

N∑n=1

(1

1 + ri

)n

[λniQαi ((1 + γnif )(1 − Rn)Xi − C ′′

ni) + R′n]. (12)

Subject to Equations (6)–(10).The quantity of work, Qα

i , in the objective function represents the intervals corresponding todifferent α-cuts of fuzzy number Q.

Qα = �Qα, Qα� = �Q − (1 − α)m, Q + (1 − α)n� (13)

If 0 ≤ t0, t1 ≤ 1 are assumed to be the coefficients which limit the interval Qαi , different values

of (t0, t1) correspond to different preference relations between intervals. Therefore, consideringspecific values for t0 and t1, the interval defined by Equation (13) is modified as follows:

Qαt0,t1

= [Qα + t0(Qα − Qα), Qα + t1(Q

α − Qα)]= [(Q + t0(Q − Q)) + (t0(m + n) − m)(1 − α), (Q + t1(Q − Q))

+ (t1(m + n) − m)(1 − α)] (14)

Definition 1 (Karimi et al. 2007):Considering the following equation:

f1(X) =I∑

i=1

N∑n=1

(1

1 + ri

)n

[λni(Qαi + t0(Q

αi − Qα

i ))((1 + γnif )(1 − Rn)Xi − C ′′ni) + R′

n]

f2(X) =I∑

i=1

N∑n=1

(1

1 + ri

)n

[λni(Qαi + t1(Q

αi − Qα

i ))((1 + γnif )(1 − Rn)Xi − C ′′ni) + R′

n](15)

where f1 and f2 are the present value of profit assuming upper and lower bounds of intervalpresented by Equation (13) as quantities of works.

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Point X ≥ 0 is called a (t0, t1)—optimum solution of problem (12) if there is no X ′ ≥ 0 whichsatisfies all the constraints as well as inequalities (16) and (17) with at least one strict inequality:

f1(X′) ≥ f1(X) (16)

f2(X′) ≥ f2(X) (17)

For a specific t0 and t1, finding such X which satisfies Equation (16) regardless of Equation (17),corresponds to the optimum decision variable (X) for the objective function which maximizesf 1(X) The same will occur when Equation (17) is considered regardless of Equation (16) whilethe objective function is to maximize f 2(X).

Therefore, considering both equations leads the decision maker to solve a bi-objective problemwith the objective functions f1 and f2.

Considering Equation (14) and definition 1, the unbalanced bidding model (Equation (12)) ismodified as follows:

I∑i=1

N∑n=1

(1

1 + ri

)n

[λni[(Q + t0(Q − Q)) + (t0(m + n) − m)(1 − α)]

× ((1 + γnif )(1 − Rn)Xi − C ′′ni) + R′

n] → Max

I∑i=1

N∑n=1

(1

1 + ri

)n

[λni[(Q + t1(Q − Q)) + (t1(m + n) − m)(1 − α)]

× ((1 + γnif )(1 − Rn)Xi − C ′′ni) + R′

n] → Max

(18)

Subject to Equations (6)–(9).If the quantities of works are considered to be asymmetric triangular fuzzy numbers, then

m = kLQ

n = kRQ

Q = Q = Q

(19)

where Q = centre values of the fuzzy numbers, kL = the fuzziness ratio on the left side and kR = thefuzziness ratio on the right side. The following consideration of preference relation betweenintervals r = [rL, rR] and s = [sL, sR] are presented for simplicity (Karimi et al. 2007):

s ≤ r ⇐⇒ sL ≤ rL ∧ sR ≤ rR

s < r ⇐⇒ s ≤ r ∧ s �= r(20)

In fact, in the above consideration, upper and lower bounds of the intervals are determinant whichcorrespond to Equations (18) with t0 = 0 and t1 = 1 (Chanas and Kuchta 1996). Substituting t0 = 0and t1 = 1, the FLP model of unbalanced bidding with asymmetric triangular fuzzy quantities ofworks (Equation (19)) will be equivalent to:

Max ZL =I∑

i=1

N∑n=1

(1

1 + ri

)n

[λni [(1 − kLi(1 − α)) Qi]((1 + γnif )(1 − Rn)Xi − C ′′ni) + R′

n]

Max ZR =I∑

i=1

N∑n=1

(1

1 + ri

)n

[λni [(1 + kRi(1 − α)) Qi]((1 + γnif )(1 − Rn)Xi − C ′′ni) + R′

n](21)

Subject to Equations (6)–(9).

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Where ZL and ZR = alternative crisp objective functions which represent the lower and upperbounds of total present value of profit for a certain α cut, respectively. In a general statement, theobjective function can be defined as:

Zt =I∑

i=1

N∑n=1

(1

1 + ri

)n

[λni [(1 + (2t − 1)(1 − α)ki) Qi]((1 + γnif )(1 − Rn)Xi − C ′′ni) + R′

n]{

ki = kLi 0 ≤ t < 0.5

ki = kRi 0.5 < t ≤ 1

(22)

where t = 0.5 results in the non-fuzzy objective function for the unbalanced bidding, t = 0 convertsthe equation to ZL with k = kL and t = 1 corresponds to ZR where k = kR.

Assuming unit price limitations as fuzzy numbers with symmetric triangular membershipshapes, Equations (7) and (8) are modified as follows (Karimi et al. 2007):

Xi ≤ Ui − eiUi(1 − α)(2q − 1) (23)

Xi ≥ Li + eiLi(1 − α)(2q − 1) (24)

In which ei = fuzziness ratio, q = degree of inequality holding true (Tanaka et al. 1985) whichranges from zero to one. The value of q is similar to the confidence level in a chance con-straint model in which probability distributions are assigned to random parameters of the modelsconstraints (Karimi et al. 2007). For q = 0.5 or α = 1 both constraints become crisp with unit pricelimitations equal to centre values of the fuzzy numbers.

As mentioned before, the aim of applying Equations (23) and (24) is to limit proposed unitprices in order to minimize the risk of rejection by the owner. For 0 ≤q ≤ 0.5 in Equation (23)the upper bound for a unit price becomes larger than the centre value of its fuzzy number whilein Equation (24) the lower bound takes a smaller amount than its centre value. Therefore, someresultant unit prices may deviate more from the base prices obtained by applying balanced bidstrategy than the case in which these bounds are assumed to be deterministic. Hence, in order toprovide higher confidence, only the values of q equal to or greater than 0.5 may be taken intoaccount and the feasible zone for the objective function shall be bounded between these two limits.It would suffice to solve the problem for q = 0.5 and q = 1, separately, which represents an FLPmodel with fuzzy coefficients in objective function and crisp constraints.

In order to deal with bi-objective programming, the constrained method is employed here (Ko1989). In this method, an idealistic interval is defined for each objective function whose optimumvalue is selected as the upper bound. In order to find the idealistic intervals, the unbalanced biddingmodel is solved at different α-cuts for different combinations of q = 0.5, q = 1 and t = 0, t = 1(Equations (6), (9), (22), (23) and (24)). Therefore, the model must be solved four times. AssumeXl

q=0.5/1 and XRq=0.5/1 are the sets of decision variables which optimize ZL and ZR at a certain

α-cut with q = 0.5/1, respectively. Substituting XRq=0.5/1/Xl

q=0.5/1 in the objective functions ofEquation (21), four values for ZL/ZR will be obtained. Plotting these eight values for each α-cutresults in four membership shapes (i.e. two membership shapes for each q). The upper and lowerbounds (ZR

min, ZRmax, Z

Lmin and ZL

max) of the idealistic interval for each objective function are alsochosen from these four values for ZL/ZR. For this purpose, two different approaches are used inthis article. As mentioned before, there are four values for each ZL and ZR at any α cut, whichare obtained from four membership shapes as defined with the objective functions of Equations(21). In the first approach, the maximum and minimum of these values (ZL and ZR) are selectedto define values of Z

R/L

max / min.

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In the second approach four fuzzy membership shapes are ranked using the Hamming Distancemethod at each α-cut and the largest and smallest ones are applied as Z

R/L

max / min.Finding Pareto optimal solutions for the bi-objective problem (Equation (21)) at each α cut is

significantly time-consuming and difficult, if not impossible. In order to reduce computationaleffort for finding non-dominated solutions, two linear membership functions μ(ZR) and μ(ZL)are defined.

μ(ZR) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1 ZR ≥ ZRmax

ZR − ZRmin

ZRmax − ZR

min

ZRmin ≤ ZR ≤ ZR

max

0 ZR ≤ ZRmin

(25)

μ(Zl) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1 ZL ≥ ZLmax

ZL − ZLmin

ZLmax − ZL

min

ZLmin ≤ ZL ≤ ZL

max

0 ZL ≤ ZLmin

(26)

The fuzzy model which maximizes the trade-off between maximum values of ZL and ZR is asfollows (Karimi et al. 2007):

Max =T

q = 0.5

ZL(x) =I∑

i=1

N∑n=1

(1

1 + ri

)n[

λni [(1 − kLi(1 − α)) Qi]((1 + γnif )(1 − Rn)Xi − C ′′

ni) + R′n

]≥ ZL

min + T (ZLmax − ZLl

min)

ZR(x) =I∑

i=1

N∑n=1

(1

1 + ri

)n[

λni [(1 + kRi(1 − α)) Qi]((1 + γnif )(1 − Rn)Xi − C ′′

ni) + R′n

]≥ ZR

min + T (ZRmax − ZR

min)

n∑i=1

Q′iXi = B

Xi ≤ Ui − eiUi(1 − α)(2q − 1)

Xi ≥ Li + eiLi(1 − α)(2q − 1)

Xa − Xb ≥ 0

(27)

where T is the trade-off rate between two objective functions which indicates the highest degreeof membership for μ(ZR) and μ(ZL). In fact, the aim of this model is to find the unit prices in sucha way that the amount of both objective functions close to their own optimum values as much aspossible while they still belong to their idealistic interval.

The traded-off membership shape of the objective functions, for different values of α cut, maynow be obtained by plotting values of ZL and ZR, resulted from solution to the model defined byEquation (27).

To provide the decision maker with optimum crisp values, this traded-off membership shapemust be defuzzified. From a number of available defuzzification methods, the centroid defuzzifieris used here and the present value of profit (ZG) as well as its related α-cut (αG) are identified.

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Although the amount of present value of profit and its α-cut are determined by defuzzificationmethods, the unit prices corresponded to the centroid remain unknown.

For different values of t in Equation (22), there exist distinct traded-off membership shapes. Infact, the centroid is located on one of them with a specific amount of t which is called tG. Therefore,to determine the unit prices related to the centroid, a combination of tG and X ={X1, X2, . . . , XI}should be found whose corresponding present value of profit equals to that of centroid (ZG) andits trade-off rate closes to trade-off rate (TG) which is identified by solving problem (24) whileα =αG as much as possible.

The model for determining unit prices corresponded to the centroid is as follows (Karimi et al.2007):

MIN

∣∣∣∣ (ZtG)traded−off − B

A − B− TG

∣∣∣∣α = αG

q = 0.5

A = max{(ZtG)Max ZR

q=0.5 , (ZtG)Max ZR

q=1 , (ZtG)Max ZL

q=0.5 , (ZtG)Max ZL

q=1 }B = min{(ZtG)Max ZR

q=0.5 , (ZtG)Max ZR

q=1 , (ZtG)Max ZL

q=0.5 , (ZtG)Max ZL

q=1 }

(ZtG)Max ZL

q=0.5 =I∑

i=1

N∑n=1

(1

1 + ri

)n[

λni [(1 + (2tG − 1)(1 − α)ki) Qi]((1 + γnif )(1 − Rn)(Xi)

Lq=0.5 − C ′′

ni) + R′n

]

(ZtG)Max ZL

q=1 =I∑

i=1

N∑n=1

(1

1 + ri

)n [λni [(1 + (2tG − 1)(1 − α)ki) Qi]

((1 + γnif )(1 − Rn)(Xi)Lq=1 − C ′′

ni) + R′n

]

(ZtG)Max ZR

q=0.5 =I∑

i=1

N∑n=1

(1

1 + ri

)n[

λni [(1 + (2tG − 1)(1 − α)ki) Qi]((1 + γnif )(1 − Rn)(Xi)

Rq=0.5 − C ′′

ni) + R′n

]

(ZtG)Max ZR

q=1 =I∑

i=1

N∑n=1

(1

1 + ri

)n [λni [(1 + (2tG − 1)(1 − α)ki) Qi]

((1 + γnif )(1 − Rn)(Xi)Rq=1 − C ′′

ni) + R′n

]

(ZtG)traded-off =I∑

i=1

N∑n=1

(1

1 + ri

)n [λni [(1 + (2tG − 1)(1 − α)ki) Qi]

((1 + γnif )(1 − Rn)Xi − C ′′ni) + R′

n

]

(ZtG)traded-off = ZG

n∑i=1

Q′iXi = B

Xi ≤ Ui − eiUi(1 − α)(2q − 1)

Xi ≥ Li + eiLi(1 − α)(2q − 1)

Xa − Xb ≥ 0

(28)

where Xi = unit price of item i related to the centroid (decision variable), tG = amount of t relatedto the centroid (decision variable), (Xi)

Rq=0.5 is the solution obtained from Equation (22) subject to

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Equations (6)–(9) at α =αG, q = 0.5 and t = 1. For (Xi)Rq=1, (Xi)

Lq=0.5, (Xi)

Lq=1 the same process

is used. When the maximum-minimum approach is used for determining ZRmin, Z

Rmax, Z

Lmin and

ZLmax in the previous step (Equations (27)), maximum and minimum values (A and B) are chosen

from four membership shapes at any α-cut in order to obtain the optimum crisp values of thedecision variables from the set of Equation (28). If the second approach (ranking fuzzy numbers)is applied, at each α-cut only those two membership functions which were ranked as largest andsmallest ones in the previous step are considered for determining parameters A and B. As it isobvious, by substituting t = 0.5 in Equation (22), the present value of profit for a non-fuzzy modelis obtained. Therefore, fuzziness ratio ki will be considered equal to kLi in the set of Equations(28) if the amount of ZG is less than the present value of profit of the non-fuzzy model (t = 0.5)and will be considered equal to kRiwhen it is more than Zα=1.

4. Model application

In general, for a balanced bid the contractor determines the total job cost, adds the markup andprorates it among items of the proposal (Table 1).

No matter how the quantities of works are estimated, they are subject to various uncertainties.The contractor may have his own estimation with range of possible values. Therefore, he may usefuzzy numbers to account for the uncertainties involved. Triangular fuzzy numbers are assumedin this study. In a real bidding condition, unit prices for some items cannot be lower than someother items. As an example, unit price for excavation in soft soil (Xee) may not exceed that forhighly consolidated one (Xre). As an illustration, it is assumed that unit price for item 3 shouldnot be less than unit price for item 2.

Since the proposed model intends to account for uncertainties in quantities of works and unitprice limitations, some assumptions are made to simplify the model without loss of generality.These assumptions are as follows:

Rn = 0, γni = 0,

Ai =N∑

n=1

(1

1 + ri

)n

λni

Input data needed for solution of the proposed FLP model are presented in Table 2.

Table 1. Estimated quantities of works and typical bid preparations.

Item number Unit cost in dollars Proposal quantity Cost in dollars Unit price in dollars Price in dollars1 2 3 4 5 6

1 80,000.00 2 160,000 90,048.65 180,097.32 300.00 2000 600,000 337.68 675,364.93 500.00 300 150,000 562.80 168,841.24 400.00 500 200,000 450.24 225,121.65 74,000.00 1 74,000 83,295.00 83,295.0

Total cost = 1, 184, 000 Total bid = 1, 332, 720.0

Column 5 (item price for balanced bid) = column 2 ∗($1, 332, 720/1, 184, 000).Subtotal of item costs ($1,184,000) plus mobilization/demobilization costs ($50, 000) = $1, 234, 000 in direct costs. Direct costs plus 8%markup ($98, 720) = $1, 332, 720 (total bid).

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Figure 2. (a) Plot of ZRmin, ZR

max, ZLmin and ZL

max and the traded-off membership shape for case example 1 using fuzzyranking method. (b) Plot of ZR

min, ZRmax, ZL

min and ZLmax and the traded-off membership shape for case example 1 using

max–min method.

Table 2. Input data to the model.

Present worth Lower limit of Upper limit of Fuzziness ratio forItem number coefficient (A) unit price (L) unit price (U) unit price limitation (e)

1 0.90 80,000.00 110,000 0.152 0.95 300.00 700 0.103 0.95 500.00 1000 0.154 0.95 400.00 900 0.155 0.90 74,000.00 95,000 0.10

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The present value of fixed cost is considered to be $30,000 which is subtracted from the presentvalue of profit.

To test the performance of the proposed modelling scheme, three cases are considered. In thefirst case, fuzzy symmetric triangular quantities of works (kL = kR) are assumed. The second case

Table 3. Left and right fuzziness ratios for the case examples.

Case example 1 Case example 2 Case example 3

Item number KL KR KL KR KL KR

1 0.0 0.0 0.3 0.0 0.0 0.32 0.2 0.2 0.7 0.2 0.2 0.73 0.1 0.1 0.5 0.1 0.1 0.54 0.2 0.2 0.6 0.2 0.2 0.65 0.2 0.2 0.8 0.2 0.2 0.8


max, ZLmin and ZL


min, ZRmax, ZL


max–min method.

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example assumes that the left fuzziness ratios of quantities of works are greater than the right ones(i.e. kL > kR) and the third one is solved when kL < kR. The centre value of each fuzzy quantityof work is equal to the quantity of work of that item proposed by the client (column 3, Table 1).The amounts of fuzziness ratios are presented in Table 3.

The model was solved for α cuts ranging from 0 to 1 and q = 0.5 and 1. Note that q = 0.5reduces the problem to an FLP model with fuzzy objective function and non-fuzzy constraints.

Both previously outlined approaches are applied to determine ZRmin, ZR

max, ZLmin and ZL

max whichare required to solve problem (27). At any α cut, the largest and smallest membership shapes areobtained using the maximum and minimum envelope method and the Hamming Distance method.

The membership shapes of ZR, ZL and Ztraded−off for each case study are shown in Figures 2,3 and 4.


max, ZLmin and ZL


min, ZRmax, ZL


max–min method.

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Table 4. Results of unbalanced bidding model with fuzzy quantities of works and fuzzy unit prices limitations.

Case example 1 Case example 2 Case example 3

Non-fuzzy Fuzzy ranking Max-min Fuzzy ranking Max-min Fuzzy ranking Max-minVariable method approach approach approach approach approach approach

α 1 0.350 0.350 0.342 0.341 0.344 0.344T 1 0.616 0.616 0.638 0.638 0.597 0.597tG 0.5 0.456 0.457 0.324 0.324 0.701 0.701z 111,284.00 111,270.60 111,283.80 92,435.20 92,404.84 131,822.80 131,820.468q1 2.00 2.00 2.00 1.86 1.86 2.16 2.16q2 2,000.00 1,977.08 1,977.68 1,675.33 1,672.28 2,368.81 2,368.77q3 300 298.28 298.33 265.21 264.89 339,52 339.51q4 500 494.27 494.42 430,43 429.77 579.03 579.02q5 1 0.99 0.99 0.81 0.81 1.21 1.21x1 80,000.00 84,666.90 85,628.84 110,000.00 110,000.00 89,417.08 95,477.45x2 300.00 300.25 335.1487 323.9051 323.9283 313.3619 348.3826x3 995.73 501.205 684.2142 625.965 625.8063 504.1805 500x4 400.00 635.0495 400 402.8577 402.86 603.8159 400x5 74,000.00 95,000.00 85,900.71 75,691.52 75,691.58 74,000.00 95,000.00

In the first case example (symmetric membership shape of quantities of work), the membershipshape of the objective function (ZL/R

traded−off ) resulting from both approaches is about symmetric.As expected, in the second case example, the left fuzziness ratio in the traded-off membershipshape is more than the right one which makes the amount of profit related to the centroid lessthan the profit calculated from the deterministic model. In the third case example, as the rightfuzziness ratio of membership shapes of quantities of works are larger than the left one, the rightfuzziness ratio in the membership shape of the objective function became larger than the left one.

By de-fuzzifying the traded-off membership shapes and solving problem (28) for each caseexample, the crisp present values of profits and their related decision variables are obtained. Theresults are shown in Table 4.

In both approaches at most α-cuts, maximum and minimum objective values (ZRmin, Z

Rmax, Z

Lmin

and ZLmax) are selected from membership shapes with q = 0.5 which corresponds to an FLP model

with fuzzy objective function and non-fuzzy constraints. Therefore, αG and TG obtained from twodifferent approaches became identical for each case example.

As it is expected, for those present value of profits which are greater than the present value ofprofit at α = 1, the obtained amount of tG is greater than 0.5 while for those which are less thanpresent value of profit at α = 1, tG becomes less than 0.5. Consequently, the amount of quantitiesof works related to the centroid are greater than their deterministic one (α = 1) when tG is greaterthan 0.5 and will be less, if tG < 0.5.

The present value of profit and quantities of works obtained from the two different approachesare about identical for the given case example. It may be concluded that, for all practical purposes,the turnover of the two approaches are almost identical and the minor differences might bedisregarded.

As the value of the profit is obtained by using the centroid de-fuzzifier approach, positiveor negative change in the profit depends on the objective function membership shape which isaffected by various elements. For instance the effect of quantities of works membership shape canbe easily observed by comparing the results of the three case examples. Project schedule, interestrate and unit price boundaries can affect the objective membership shape as well.

Therefore, in some cases the profit may decrease or increase, compared to the results of thedeterministic model such as case examples 2 and 3, respectively. The main objective in consideringuncertainties in the unbalanced bidding model is to maximize the present value of profit under

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these uncertainties which may not necessarily lead to a profit higher or lower than that of thedeterministic model.

5. Conclusion

Some of the existing unbalanced bidding models assume that any effective parameter such asquantities of works, is certain and deterministic. However, in real world problems exact valuesfor some of these quantities are not precisely known. Although there can be substantial risk due touncertainties involved, the unbalanced bidding strategy can help the contractor to submit a lowerbid or improve the profitability of his tender. This article presented an FLP model of unbalancedbidding considering some uncertainties such as quantities of works and unit price limitations,while maximizing the present value of profit.

Two slightly different approaches were developed to consider uncertainties of unit price lim-itations. The proposed modelling approach was applied to three previously examined and/orhypothetical case examples with fuzzy work quantities and/or fuzzy unit price constraints. Theresults of these comprehensive fuzzy models were compared with those of the deterministicmodelling approach. It was shown that, for the cases under consideration, the turnover of twoapproaches is about identical.

In comparison to the LP method, the presented FLP method incorporated all possible situationswhich make the proposed approach more realistic and robust. When uncertainties are consideredthrough the fuzzy modelling approach, the present value of profit for these case examples increasedor decreased, which seems to be case dependent. Employing the proposed model and its results,the contractors might have a better understanding of the effects of uncertainties on work quantitiesand the ranges of the proposed unit prices on their overall earning.

References

Afshar, A. and Amiri, H., 2008. A fuzzy based model for unbalanced bidding in construction, 1st international conferenceon construction in developing countries (ICCIDC–I), 4–5 August, Pakistan. Karachi: NED University of Engineeringand Technology, 57–63.

Bellman, R.E. and Zadeh, L.A., 1970. Decision-making in a fuzzy environment. Management Science, 17 (4), 141–146.Cattell, D.W., 1984.A model for item price loading by building contractors. Department of Quantity Surveying: University

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Plenum Press.Gates, M., 1959. Aspects of competitive bidding. Annual Report. Connecticut Society of Civil Engineers.Gates, M., 1967. Bidding strategies and probabilities. Journal of Construction Division, 93 (C01), 75–107.Green, S.D., 1989. Tendering: Optimization and rationality. Construction Management and Economics, 7, 53–63.Karimi, S., Mousavi, J., Kaveh,A., andAfshar,A., 2007. Fuzzy optimization model for earthwork allocation with imprecise

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risk-based approach to unbalanced bidding in construction projects

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