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Unformatted text preview: Construction Management and Economics, 1992, 10, 69—80 Unbalanced bidding on contracts with
variation trends in clientprovided quantities TONG YIZHE and LU YOUJIE
Department of Civil Engineering, Tsinghua University, Beijing, China This paper examines unbalanced contract bidding, a strategy for the allocation of rates to unit quantities for the beneﬁt of the bidder. A mathematical model is proposed which attempts to objectively exploit
variation trends in clientprovided quantities. It is shown that the model can be solved by two
methods — linear programming and the maximum—minimum method. The maximum—minimum method is
preferred for most realworld situations. Keywords: bidding strategies, unbalanced bidding, cash ﬂow, operations research Introduction Unbalanced bidding is concerned with the judicious allocation of rates to unit quantities in
order to provide some increased beneﬁt to one of the parties involved, usually the bidder. It is
sometimes used in (Tong, 1989): (a) Unitprice contracts with variation trends in clientprovided quantities. (b) Both unitprice and lumpsum contracts by judiciously increasing the unit price of certain
items in order to inﬂate progress payments in the early stages of the contract and, at the
same time, decreasing the unit price of other items to keep the total bid at a competitive
level. (c) Staged or phased projects with a decreased bid in the ﬁrst stage to improve the
competitive power, followed by an increased bid in succeeding stages to make up the
losses incurred in the ﬁrst stage. Trialanderror methods may be used for (b), and (c) has been shown to be essentially a
problem concerned with securing the latter stage contracts by strategic means as yet without
any mathematical treatment (Tong, 1988). Stark (1968) analysed the situation in which the bidder is interested in maximizing the
present worth of a set of unit prices combined into a total bid price, subject to (1) bid
constraints, e.g. the total bid price ﬁxed in advance, and (2) unit rate constraints, e.g. for rock
excavation rates to exceed those for earth excavation, rates to have maxima/minima values.
Stark also examined the extension of this work to cases where the bidder had detected errors
in clientprovided quantities, proposing a linear programming method for its solution. Hughes (1982) has subsequently treated unbalancing as two distinct activities: ( 1) finance
cost/cashﬂow unbalancing, i.e. item rate manipulation to take into account ﬁnance cost and 0144—6193/92 $03.00+ .12 © 1992 E. & F.N. Spon Copvriqht © 2001 . All Rights Reseved. 70 Tong and Lu cashﬂow considerations; and (2) error exploitation unbalancing, i.e. item rate manipulation
to exploit errors in clientprovided quantities. Green (1986) investigated the extent to which these two types of unbalancing were used in
practice in a series of interviews involving three contractors and three consultants. Two of the
contractors used cashﬂow unbalancing and considered the practice to be widespread,
contrary to the consultants’ views. None of the contractors unbalanced for ﬁnance cost
purposes. Error exploitation unbalancing, however, was used by all the contractors, the
opportunities for the practice being quite common because of ‘blatant mistakes’ in the bid
documents. Apparently, the contractors’ estimators detected the errors and subjectively
adjusted the associated rate accordingly, the total bid value being maintained by an equal and
opposite adjustment in the ‘preliminaries’ section of the documents. In a later paper, Green (1989) suggests that the current crude redistribution of values to the
preliminaries could be replaced by a more sophisticated treatment with the use of a
computerbased estimating system. The argument is made, however, that error exploitation
unbalancing generally would be difﬁcult to apply‘. . . in a systematic and optimizing method
. . .’ (Green, 1989, p. 57). In this paper, we describe a method for error exploitation unbalancing that is both
systematic and optimizing, relying on little more than the estimator’s current ability to detect
errors. First, the situation is described in which the bidder has alternative quantity estimates
for two work items. This is formulated as an optimization problem with two constraints « the
total bid and range of unit price — from which a solution may be derived by linear
programming methods. Extending this technique to a full bill of quantities, however, is
difficult i if not impossible 7 due to the complexities involved. To overcome this difficulty, a
second method, the maximum—minimum method, is proposed, which uses an iterative
approach to locate the solution. Finally, an example is provided to illustrate the use of the
new method for error exploitation unbalancing. Bidding on unit price contracts with variation trends in clientprovided quantities An important factor considered by contractors before deciding upon a unit rate is the
reliability of the quantity stated in the bills of quantities (Hughes, 1982). Although the
requirement to make judgements on the reliability of stated quantities is more critical in civil
engineering work, where bills consist of ‘approximate’ quantities there are some commonly
used building contracts which also allow the use of approximate quantities, and even bills of
ﬁrm quantities usually contain quantities of a provisional nature (Green, 1986). A contractor
detecting ‘errors’ resulting from unreliable quantities in the bills at the time of bidding may be
able to gain extra proﬁts by judicious rate manipulation or unbalancing (Green, 1989). The
following example illustrates the concept involved. Example Consider a unit price contract with quantity errors in unit prices as shown in Table 1. The
lump sum price of the two items based on their normal unit prices is: S=3000 m3 x 40 $/m3 +2000 m3 x 30 $/m3 =$180 000 Unbalanced contract bidding 71 Table 1. Quantity errors in two items
———— Quantity (m3) Work a Unit price
item by client by bidder ($/m3) A 3000 3800 40 (UPA)"
B 2000 1500 30 (UPB)“ ——__——
“UPA and UPB denote the unit price of items A and B respectively. Unbalanced bidding Progress payments are usually based on the actual quantities of each item. Thus, if the bidder
uses a markup value of UPA’ for item A instead of UPA, he should receive more income than
when bidding normally. At the same time, the bidder should consider two constraints. 1. Bid constraint. The total bid price needs to remain competitive, i.e. the combined price of
A and B shall be approximately equal to $180 000. Hence UPB should be reduced to
accordingly to UPB’. 2. Range of unit price. The range of the variation of UPA and UPB should not be too wide.
Supposing UPA'= UPA x 110%
UPB’=UPB x 90% then the new combined bid, S’, is given by
S’ = 3000(40) (110/100)+ 2000(30) (90/100) = 186 000
But
S’ — S = 6000 and this difference needs to be eliminated. Letting UPB’ be 30(90/ 100) = 27, and UPA’ be X,
then 3000X+ 2000(27) = 3000(40) + 2000(30)
which gives
X = 42 Thus the bid constraint can be satisﬁed. The unbalanced bid and expected income are shown
in Table 2. Therefore, the expected extra income is: 200100—180 000:20100 Copvriqht © 2001 . All Rights Reseved. 72 Tong and Lu Table 2. Unbalanced bid and expected income Data in bid Data as expected
Work
item Quantity (m3) Unit price Total Quantity (m3) Unit price Total
A 3000 42.00 126 000 3800 42.00 159 000
B 2000 27.00 54 000 1500 27.00 40 500
180 000 200 100 __—_________————————————__—______ In practice, the main points of this type of unbalanced bidding are: 1. It is assumed that accurate quantities may be estimated by the bidder. If the bidder’s
estimate of the quantities is incorrect, he may suffer losses by using this approach. In the
above example, let the actual quantity of item A be Y, and consider the inequality: 42.00 Y+ 1500(27.00) < 3000(40.00) + 2000(30.00)
namely Y< 3085.7 m3, in which case the contractor would suffer a loss. It is likely, however, that inaccuracies in item rate estimates will far exceed inaccuracies in
quantity estimates, making the risks involved in the method of minimal consequence and
obviating the need for an explicit risk representation. 2. The range of unit price variation for unabalanced bidding should not be so wide that the
client would reject the bid. This is clearly a matter of judgement for the bidder. The example
assumes the acceptable range to be i 10%, but it is likely that up to i 5% could be used on
occasions (cf. Beeston, 1975, reproduced in Green, 1989, Table 2). 3. The trialanderror method used in the example easily locates the solution if (a) the
number of relevant work items is not too large, and (b) the expected extra income is not too
high. Otherwise, an accurate method is necessary to obtain an optimal solution. This is
presented below. Mathematical formulation Let q, (i = 1, 2, . . . , n) be the clientprovided work item quantities involved in the unbalanced
bidding, and denote the vector Q as: Q:(q1v ‘12, ' ' ‘ a qn)T Unbalanced contract bidding 73 Let ai (i=1, 2, . . . , n) denote the expected quantities corresponding to qi, their values
calculated by the bidder, and denote the vector A as: A=(a1, a2, . . . , an)T Denote p1, yi, li, ui as follows: pi is the normal unit price of qi, and the vector P is given by: P=(pi,p2,    ,pn)T yi is the undetermined bid unit price of related items. Similarly, the vector Y is given by: Y=(y1,y2,  ~  ,yn)T ui is the upper bound of yi, determined by the bidder, and
U=(u,, uz, . . . , un)T
II. is the lower bound of yi, determined by the bidder, and L=(ll, lz, . . . , In)T Sometimes, there would be very few items with either decreasing or increasing expected
quantities. In this case, some items without any expected variation may be considered, and
the expected extra income maximized. Let these quantities be denoted as qi (i = s +1, 3 +2, . . , t), which should be chosen from items either in the early or late stages of construction (termed ‘early’ and ‘late’ items here). Then deﬁne [—i = ai/qi where F1 21—2 2 . . . > F". this means that the order shall be arranged
anew as follows: Fi>1 (i=1,2,...,s)
[1:1 (i=s+l,s+2,...,t)
ﬁ<1 (i=t+l,t+2,...,n) The problem then becomes max <D(Y)=ATY—QTP (1) subject to
QTY: QTP (2)
Y s U (3) Copvriqht © 2001 . All Rights Reseved. 74 Tang and Lu YzL (4) Evidently, the above formulation is a ‘linear programming’ problem and an optimal solution
can be derived by using the Simplex Method.
Deﬁning X=Y—L, and rewriting Equations 14 with X instead of Y~—L, we obtain: max f(X) = ATx + ATL — QTP (5)
QTX = QT(P* L) (6)
XSU—L (D X 2 O (8) Obviously, the number of constraint equations is n+1, which is a great deal less than the
2n+1 constraints in (2), (3) and (4); hence the amount of calculation involved when using
(5)—(8) will be reduced accordingly. The complexity of solving the problem by the Simplex Method is rather great, especially
when n is very large. A new method was therefore developed by the authors to simplify the
procedure. This is called the maximumrminimum method. A new procedure: the maximum—minimum method Assuming yizui (i=l,2,...,s) yi=li (i=s+1,s+2,...,t,t+1,t+2,...,n) enables Equations 3 and 4 or 7 and 8 to be satisﬁed. Equations 2 and 6, however, may not be
satisﬁed at this point.
Consider the lefthand side of Equation 6: QTX = QT(Y — L) =<Zqiui+ Z qili>—<Zqili+ Z qili)
i=1 t=l t=s+i t=s+1
S
= Z qi(ui_li)
:21 Let S denote this result.
The righthand side of Equation 6 can be written as: ,, QQMdmm—Mﬁﬂhmm—m '_ Unbalanced contract bidding QT(P—L)= Z [IiPi_ Z qili
i=1 t=l Let R denote this result.
Deﬁning A=R—S, there may be three cases for A, namely: A>O A20 A<0 75 To solve the problem, it is necessary to calculate A ﬁrst, then take one of the following steps according to the value of A. Step 1: A>0 In this case, Equation 6 is not satisﬁed. Thus let an early item x, with [—= 1, say xs+ 1, be increased from 0 to
Xs+1 =A/‘Is+1 = (R_S)/qs+l
namely: Ys+l=A/qs+1+ls+1 If
Als+1S Us+1 _ls+l
then xs+ 1 satisﬁes Equation 7 and the optimal solution is obtained, i.e. X"‘=(u,—l1 ,...,us—ls,xs+1,0,...,0)T OI‘ *_ T
Y —(u1,...,us,xs+1+ls+1,ls+2,...,l,,..., In). On the contrary, if xs+1>us+1_ls+l then it is required that xs+1=us+l—ls+1 and let another early item xi, say x5”, be increased to Copvriqht © 2001 . All Rights Reseved. (10) 76 Tong and Lu xs+2=1/q5+2[A_(us+1—ls+1)qs+1] (11)
If xx+2 satisﬁes Equation 7, the optimal solution is obtained as
X*=(u1~ll, . . . ,uS—ls,us+1—ls+1,x5+2, 0, . . . 0)T (12) otherwise, repeat the same step until every xi satisﬁes Equation 7, and Equation 6 is satisﬁed. Step 2: A=0 In this case, all the constraint Equations 6—8 are satisﬁed. The optimal solution is: X*=(u1~ll,...,u —ls,0,...,O)T Y*=(u1,...,u 0,...,0)T (13) Step 3: A<0 In this case, Equation 6 is not satisﬁed. Thus a unit price with Fi> 1 is reduced. Let x5, for
example, be reduced ﬁrst: x5:A/qs+uS—ls (14)
If
x520 i.e.ys=xs+lszlS
the optimal solution is obtained as
x*=(u1#l1,...,us_1—ls_1,xs,0,...,0)T (15)
Otherwise another unit price with F> 1 is reduced as follows: xsil':l/qs—1[A+(usls)qs]+us‘ls1 If xs_ 1 20, the optimal solution is obtained, otherwise it is necessary to repeat the above
process until Equation 7 is satisﬁed before the optimal solution is obtained.
To sum up, the unbalanced bidding method requires the following steps: 1. Check the quantities of work items given in the client—provided bills of quantities, paying Unbalanced contract bidding 77 attention to earth work, pipe work, other outside work and works with large quantities.
Then, ﬁnd out all of the items with variation. 2. Determine the normal unit price pi of these items and their corresponding upper and lower
bounds, ui and 1,. In general, consider ulglin
[£20.9pi 3. Make a trial calculation of the total price, then adopt one of the following measures:
(a) Compute the value of A on condition that Equation 6 or 2 could be approximately
satisﬁed and then take corresponding steps to obtain the optimal solution.
(b) On condition that the number of items with [—1. < 1 is small, consider some work items with l'i=l in the calcluation. Denote these items by qi, i=s+1, s+2, . . . , 1?.
Compute the value of A and adopt the corresponding method to obtain the optimal
solution. Example Find the optimal unbalanced bidding solution for a unitprice contract with clientprovided
quantities of work items qi and unit prices pl. as follows: qi=(15 000,12 000, 30 000,1800,...)T
pi: (4.00, 3.00, 5.00, 6.00, . . .)T Solution 1. Check the quantities of work items according to drawings, etc., for the contract. It is found
that the expected quantities of the preceding four quantities are as follows: (1,: (20 000, 18 000, 24 000, 1000)T
2. Determine the upper and lower bounds of the above four unit prices: u,=(4.40, 3.30, 5.50, 6.60)T
li=(3.60, 2.70, 4.50, 5.40)T 3. Make a trial calculation of total bidding price, assigning u1 , uz, l3 and I4, and compare it
with normal bid price, p, to p4. [15 000(4.40)+ 12 000(3.30) + 30 000(4.50)+ 1800(5.40)] — [15 000(400) + 12 000(300)
+ 30 000(5.00)+1800(6.00)]= — 6480 Copyright © 2001 . All Rights Reseved. 78 Tang and Lu This result shows that these corrections would reduce the total price. Consider now a ﬁfth
item in the early stage of construction with its data as follows: q5 26000 m3 4. Computing the value of Fi=ai/qi and arranging Fl. in decreasing order, gives: 18 000 20 000 6000
=——=1.5 =——=1.33 =—~=1.0
Fl 12000 D 15 000 F3 6000
24000 1000
r4 30000 is 1800
qi=(q1,q2,~,qs)T = (12 000, 15 000, 6000, 30 000, 1800)T
14,: (3.30, 4.40, 11.00, 5.50, 6.60)T
1,: [2.70, 3.60, 9.00, 4.50, 5.40)T Now compute A = R — S =12 000(3 — 2.7) + 15 000(4—3.6)+ 6000(10—9)+ 30 000(5 ~4.5)+1800(6— 5.4)
= 31680 2
S: 2 Giiuili)
i=1 =12 000(3.3—2.7)+ 15 00014.4— 3.6) :19 200
=R—S=3l 680419 200=12480>0 Thus a certain item with A: 1, say the item 5+1. should be taken into account for
compensation. By trial calculation with Equation 9, we obtain: A 4
!=1_2——89=2.08>u,—1,=2 x :
S qs 6000 Hence we take x5=2. Unbalanced contract bidding 79 Then calculate by trial another xi of another item with F1 <1, i.e. x4, which should be
solved from Equation 11: 1
x4 = — [A— (“3 _13)‘13]
‘14 =‘148 — 6
30000[2 0 2( 000)] 480 0 000
0.016, N and
x4<u4—l4=5.5—4.5=1
Thus Equation 6 is satisﬁed, the optimal solution is obtained as follows: X* = (0.6 0.8 2, 0.016, 0)T
Y* =X* +1
= (3.30 4.40 11.00 4.516 5.40)T Therefore, the extra income would be max <1) (Y)=A‘Y—QTP qipi Mo 5
= Z ai)’;_
i=1 i l = 3.30(18 000)+4.4(20 000)+11.00(6000)+4.516(24 000) + 5.40(1000)— 300(12 000)
—4.00(15 000)—10.00(6000)— 500(30 000)—6.00(1800) = 327184—316 800 =10 384 In this case, therefore, 316 800 is the total bidding price of relevant items with normal unit
pricing, and 327 184 is the total expected contract income.
The total expected extra income by using this method is: 327 184 — [3(18000) + 4(20 000) + 10(6000) + 5(24 000) + 6(1000)]
= 7184 Copyright © 2001 . All Rights Reseved. 80 Tang and Lu Conclusions This paper describes an approach to error exploitation unbalancing, i.e. unbalanced bidding
where clientprovided quantities contain errors identiﬁed by the bidder during the bidding
process. It is shown that the unit prices may be adjusted to gain advantage of the situation while still
retaining the competitiveness of the overall bid. Two techniques are examined to optimize
the results of this process. First, a linear programming formulation is proposed by which the
solution can be obtained by the Simplex Method. Although working well for smallscale
versions of the problem, the technique was found to be inadequate for most realworld
situations. To overcome this, a new method was developed, called the maximumiminimum
method, which utilizes an iterative method in a far more efﬁcient manner than the standard
technique. Although as yet untested, the technique is likely to be of real value in helping bidders to
capitalize more objectively on a situation that is typical in contract bidding, and also provide
an indication of the care needed in the production of clientprovided quantities. Acknowledgements The authors gratefully acknowledge the kind advice and recommendations given by an anonymous
reviewer of an earlier draft of the manuscript. References Beeston, D.T. (1975). One statisticians view of estimating, Chartered Surveyor, Building and Quantity
Surveying Quarterly, 12(4). Green, SD. (1986). The unbalancing of tenders. MSc dissertation, Department of Building, Heriot—
Watt University. Green, SD. (1989). Tendering: optimisation and rationality. Construction Management and
Economics, 7, 53763. Hughes, GA. (1982). Reliability of Civil Engineering Quantities. Construction Press. Stark, RM. (1968). Unbalanced bidding models 77 theory. Journal of the Construction Division,
Proceedings of the American Society of Civil ASCE, 94, (C02), 1977209. Tong, Y. (1988). Unbalanced bidding on project contracts. Modernization of Construction
Management, 2. Tong, Y. (1989). Project bidding from inception to completion. China Economy. ...
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