招投标平均价格

Below-Average Bidding MethodPhotios G.Ioannou,M.ASCE 1;and Rita E.Awwad 2Abstract:The low-bid method,typically used for competitive bidding in the United States,may result in a contract with a firm that submits either accidentally or deliberately an unrealistically low-bid price.Such an occurrence hurts both the owner and the contractor by promoting disputes,increased costs,and schedule delays.To address this problem,other countries have adopted bidding methods based on the average of the bids submitted.One such approach is the below-average method where the winning bid is closest to but below the average of all bids.A competitive bidding model for the below-average-bid method is presented and its merits relative to the average-bid method and the low-bid method are explored.The below-average-bid process is investigated analytically and through Monte Carlo simulation.The results of bidding models for the below-average,the average,and the low-bid methods are presented in four easy-to-use nomograms which allow contractors to determine the optimal lump-sum bid price for each method without the need for complicated analysis.A comparison of the three methods provides information and insights to help owners with the difficult choice of a suitable bidding method for the project at hand.DOI:10.1061/͑ASCE ͒CO.1943-7862.0000202CE Database subject headings:Construction management;Contracts;Bids;Pricing;Models;Taiwan;Italy;Peru;Florida .Author keywords:Construction;Contracts;Bids;Pricing;Competition;Model;Simulation;Average bid;Below-average-bid;Low bid;Markup;Contingency;Probability;Innovation;Taiwan;Italy;Peru;Florida .IntroductionThe majority of competitively bid construction contracts in the United States use the low-bid method where the contract is awarded to the firm submitting the lowest responsible -petition on price encourages contractors to adopt cost-saving technological and managerial innovations and then pass these savings on to the owner who receives a project of specified qual-ity at the lowest price.When the number of bidders is large,however,as is the case in a slow economy,an owner runs a significant risk of selecting a contractor that has either acciden-tally or deliberately submitted an unrealistically low price.A con-tractor cannot adhere to such a price and at the same time expect to complete the project according to plans and specifications and also make a reasonable profit.This often results in excessive claims and disputes during construction that lead to schedule de-lays,compromises in quality,and increased costs ͑Grogan 1992͒.The Florida DOT ͑FDOT ͒,for example,reported that,on average,traditional low-bid contracts had a 12.4%cost overrun and a 30.7%time overrun,whereas nontraditional contracts only had a 3.6%cost overrun and a 7.1%time overrun ͑Florida DOT 2000͒.In response to these problems,some countries have recognized that the lowest price may not be the best price for the owner and developed bidding procedures based on the average of the bids received.The rationale for these methods is that a price close tothe average should offer a fair price to the owner and allow the contractor to perform the work at specified quality and at a rea-sonable profit.In this paper,we present various bidding methods based on the average bid and develop a competitive bidding model for the below-average-bid method.Our objectives are to help owners with the selection of an appropriate competitive bidding method and to provide easy-to-use tools to help contractors choose an optimum markup for each of the bidding methods presented.The below-average-bid model is investigated analytically and through Monte Carlo simulation using a decision analysis approach ͑Roth-kopf 2007͒and the results are presented in four easy-to-use no-mograms that can be used by any contractor to analyze a competitive situation without the need for complicated mathemat-ics.We also present the results of bidding models for the average-bid method ͑Ioannou and Leu 1993͒and the low-bid method ͑Friedman 1956;Ioannou 1988͒,which are used to compare the three models and explore their merits relative to each other.We conclude with a discussion and summary.Average-Bid MethodsOver the years,several competitive bidding methods have been developed where the winner is determined by comparing the sub-mitted bids to their average.Different average-bid methods use different procedures for calculating the average or use different criteria for determining the winning bid.For example,some use an arithmetic average or a weighted average,while others use the average of the remaining bids after all bids that differ more than a certain percentage from the average of all other bids are elimi-nated.Similarly,the winner might be the contractor whose price is closest to the average or the contractor whose bid is closest to but below the average.The former,for example,is used in Taiwan while the latter is used in Italy.1Professor,Dept.of Civil and Environmental Engineering,Univ.of Michigan,Ann Arbor,MI 48109-2125͑corresponding author ͒.2Doctoral Candidate,Dept.of Civil and Environmental Engineering,Univ.of Michigan,Ann Arbor,MI 48109-2125.Note.This manuscript was submitted on March 30,2009;approved on January 29,2010;published online on February 8,2010.Discussion period open until February 1,2011;separate discussions must be submit-ted for individual papers.This paper is part of the Journal of Construc-tion Engineering and Management ,V ol.136,No.9,September 1,2010.©ASCE,ISSN 0733-9364/2010/9-936–946/$25.00.D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y S HE N Z H E N U N I V E R S I T Y o n 12/31/15. C o p y r i g h t A S C E .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .Other countries have adopted even more complex variations.Peru,for example,uses a procedure where if less than three bids are received,a bidding agency may award the contract to the lowest bidder.When three or more bids are received,the average of all bids and the base budget are calculated,and bids that lie 10%above and below this average are eliminated.A second av-erage of the remaining bids and the base budget is calculated,and the bid closest to but below the second average is the winner.If none of the bids lie below the second average,the winner is the bid closest to but above the average ͑Henriod and Lantran 2000͒.In 1996,at the urging of the secretary of transportation,Florida passed a statute allowing use of alternative bidding tech-niques on $60million out of the approximately $1.25billion worth of annual FDOT contracts.As part of this effort,FDOT investigated several bidding alternatives on state-funded highway projects,including the bid-averaging method .According to this method,if there are five or more bidders,the high and low bids are discarded,the remaining bids are averaged,and the bidder closest to the numerical average is awarded the contract.If there are three or four bidders,no bids are discarded,and if one or two bids are received,the job is advertised again.The objectives of this approach were to encourage contractors to submit true and reasonable bids for the work that would result in higher quality projects with less monetary disputes.The bid-averaging method was used primarily on typical low-risk projects,such as bridge rehabilitation and pavement resurfacing ͑ASCE 1998;AASHTO 2001͒.The bid-averaging method was one of the nontraditional contract methods used by FDOT to produce the cost and time overrun statistics referenced earlier ͑Florida DOT 2000͒.Average-Bid Bidding ModelThe average-bid method,where the winning bid is the one closest to the average of all submitted bids,has been analyzed in detail by Ioannou and Leu ͑1993͒.Though this method uses the simplest possible rule to determine the winner,it has proven very difficult to model analytically.As shown by Ioannou and Leu ͑1993͒a closed-form competitive bidding model can be formulated only in the simple case where a contractor faces two opponents.When the number of opponents is greater than two,the problem was solved through Monte Carlo simulation.The final results of the bidding model for the average-bid method were presented in four nomograms that allow a contractor to select an optimal markup for a lump-sum bid without the need for complicated mathematics.These results are replicated here to allow comparison between the average-bid method,the below-average-bid method,and the low-bid method.A testament to the difficulty of modeling bidding methods based on the average is the fact that the competitive bidding model of Ioannou and Leu ͑1993͒is still the only one to do so.Yet,this model has been used for research in several domains outside construction and has been referenced in diverse publica-tions over the years.Inquiries and interest from several research-ers about the availability of competitive models for other variants provided some of the impetus for the development of the below-average model presented here.Below-Average-Bid MethodIn this paper we analyze the below-average-bid method where the winning bid is the one closest to but less than the arithmeticaverage of all submitted bids.We present a competitive bidding model for the below-average-bid method and compare it to bid-ding models for the average and the low-bid methods.We close with some remarks about all three methods.We shall use the three examples shown in Fig.1to illustrate some of the subtleties of the below-average-bid method and com-pare it to the average and low-bid methods.Fig.1͑a ͒shows six contractors who have submitted the following bid prices:83,85,87,92,95,and 98.The average bid price is 90.The bid at 92is closest to the average and would win the project under the average-bid method.Under the below-average-bid method,how-ever,the winner would be the bid at 87.The low-bid winner is the bid at 83.This example was constructed so that there is symmetry in the arrangement of bid prices:three bids are above the average and three below.This is the situation visualized by most people when they are introduced to the average-bid method for the first time.Moreover,in order to distinguish between the average-bid method and the below-average-bid method,the average bid for this ex-ample ͑90͒is closer to the bid above ͑92͒,rather than the one below ͑87͒,to make it clear that the two methods can lead to different winning bids.Fig.1͑b ͒shows three contractors who have submitted the bid prices 96,97,and 99.The average bid price is 97.3.The bid at 97is clearly the closest to and below the average and would win the project under both the below-average and the average-bid meth-ods.Let us now consider a fourth contractor who is about to submit a bid.Clearly,if the fourth bid happens to be in the narrow range between 97and 97.3,it will indeed be the winner.However,this is not the only possibility.Fig.1͑c ͒shows what happens if the fourth bid is 91.In this case,the average of the four bids drops to 95.8,which is below all three original bids.Hence,under the below-average-bid method,the winner is actually the low bid price at 91.This example illustrates a very important and not so obvious property of the below-average-bid method.A contractor can win by submitting a bid so low ͑91͒that the resulting bid average ͑95.8͒is below the minimum of its competitors’bid prices ͑96͒.This characteristic makes the below-average-bid method similar to the low-bid method.This and other possibilities for a contrac-tor to win under the below-average-bid method are discussed in detail below.It is clear from these examples that both average-bid methods are significantly more complex and thus more difficult to model than the low-bid method.For the low-bid method,the main vari-able of interest,from a contractor’s perspective,is the minimum of the bid prices submitted by the other competitors.The contrac-tor’s own bid price is the decision value to which the opponents’bids are compared.The average bid price,on the other hand,depends on all bids,including the contractor’s own bid price.969797.3999092949698100(b )9890.0959283878580859095100(a )91969795.8999092949698100(c)Fig.1.Below-average-bid method examplesD o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y S HE N Z H E N U N I V E R S I T Y o n 12/31/15. C o p y r i g h t A S C E .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .Furthermore,in order to determine the winner we need to com-pare every bid to the average.A change in any of the bid amounts or the number of bidders can have a profound effect on any par-ticular bidder’s chance of winning.Finally,an important difference is that the average-bid method requires the calculation of bid differences from the average whereas the below-average-bid method does not.The below-average-bid method requires only that bids be sorted in increasing order and their average compared to each one,from the largest to the smallest.The first bid found that is less than the average is the winner.A similar procedure can indeed be followed for the average-bid method.In that case,however,it is still necessary to calculate the difference between the average and the bids imme-diately above and below the average to determine which one is closer and thus the winner.This observation has important effects on the mathematical analysis below.Basic DefinitionsThe methodology for the development of the competitive models presented here is that of decision analysis for a risk-neutral deci-sion maker as recommended by Rothkopf ͑2007͒.It should be noted that similar competitive bidding models for risk-averse de-cision makers have also been developed by the writers and will appear in future articles.The notation adopted in this paper is to indicate random vari-ables and the identity of contractors using capitalized symbols;decision variables and the values of random variables are shown using lower-case symbols.We shall examine the bidding process from the perspective of a particular contractor A 0who is bidding on a new project against n competitors A 1,A 2,...,A n .Let C i be the project cost estimate and B i be the bid price for competitor A i ,i =0,1,...,n .We shall assume that contractor A 0has already determined a cost estimate c 0and needs to decide on a bid price b 0to maximize its expected profit V .If we let the random variable C represent the uncertain cost of the project to A 0,then A 0’s expected profit isE ͓V ͉b 0,c 0͔=P ͓A 0wins ͉b 0͔͵c͑b 0−c ͒f C ͑c ͒dc ͑1͒=P ͓A 0wins ͉b 0͔͑b 0−E ͓C ͔͒͑2͒=P ͓A 0wins ͉b 0͔͑b 0−c 0͒͑3͒Thus,because A 0is assumed risk neutral,the maximization ofA 0’s expected profit requires that the cost estimate c 0be equal to the expected value of the actual cost of the project if won.Typi-cally,contractor A 0chooses a markup m 0to arrive at a bid price b 0=c 0+m 0.Equivalently,contractor A 0may select a bid-to-cost ratio x 0=1+m 0/c 0and multiply it by its cost estimate c 0to arrive at its bid price b 0b 0=c 0+m 0=͑1+m 0/c 0͒c 0=x 0c 0͑4͒For contractor A 0the values c 0,m 0,x 0,and b 0are decisionvariables even if their values have not been determined yet.In contrast,A 0does not know the competitors’cost estimates C i and bid prices B i .Given this state of information,A 0considers these to be random variables even though C i and B i are clearly decision variables for the corresponding contractor A i .Furthermore,A 0’s state of information,and hence its treatment of these variables,does not change depending on whether contractor A i has already determined and thus fixed the values of C i and B i .The ratio of contractor A i ’s bid B i to contractor A 0’s own cost estimate c 0represents the apparent bid-to-cost ratio used by op-ponent A i and will be shown as X iX i =B i c 0,i =1,2,...,n͑5͒It should be noted that for contractor A 0the ratio x 0=b 0/c 0is a decision variable,whereas the X i ͑i =1,...,n ͒are the main ran-dom variables for the bidding model analysis that follows.Standardization AssumptionsThe bidding model for the below-average-bid method is based on the same two assumptions in ͑Ioannou 1988͒,which are required for Friedman’s low-bid model ͑Friedman 1956͒,and which were also used for the average-bid model ͑Ioannou and Leu 1993͒.This allows a comparison of all three bidding models on an equal basis.The two assumptions are as follows1.In order to eliminate the effect of project size,each oppo-nent’s bid B i ͑i =1,2,...,n ͒is standardized by taking its ratio to contractor A 0’s cost estimate c 0.The resulting ratios X i ͑i =1,2,...,n ͒are assumed to be mutually independent.2.The probability distribution of each apparent bid-to-cost ratioX i ͑i =1,2,...,n ͒does not depend on the values of x 0and c 0that A 0choosesP ͓X i Ͻx i ͉x 0,c 0͔=P ͓X i Ͻx i ͔,i =1,2,...,n͑6͒The theoretical and practical ramifications of these assumptions are examined in detail by Ioannou ͑1988͒and are not presented here.Mathematical FormulationThe winning contractor under the below-average-bid method is the one whose bid is closest to but less than the average of all bids submitted.Letting the average of all bid prices be B ¯=͑b 0+B 1+¯+B n ͒/͑n +1͒,the formal definition for the event “contractor A 0wins”can be expressed in several different ways as follows͕A 0wins ͉b 0͖=͕പB i ϽB¯͓B ¯−b 0ϽB ¯−B i ͔പ͑b 0ϽB ¯͖͒͑7͒=͕പB i ϽB¯͓B i Ͻb 0͔പ͑b 0ϽB ¯͖͒͑8͒=͕പi =1n͓͑B i Ͻb 0͒ഫ͑B ¯ϽB i ͔͒പ͑b 0ϽB ¯͖͒͑9͒=͕പi =1n͓͑b 0ϽB ¯ϽB i ͒ഫ͑B i Ͻb 0ϽB ¯͔͖͒͑10͒=͕പi =1n͓͑b 0−B i ͒͑B ¯−B i ͒Ͼ0͔പ͑b 0ϽB ¯͖͒͑11͒Notice that the opposing bids B i in Eqs.͑7͒and ͑8͒are only thosethat satisfy:B i ϽB ¯.The remaining expressions involve all oppos-ing bids B i .Moreover,since B 1,B 2,...,B n are random variables,D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y S HE N Z H E N U N I V E R S I T Y o n 12/31/15. C o p y r i g h t A S C E .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .so is the average B ¯.At the same time,since b 0is a decision variable,the average B ¯is also influenced by contractor A 0.An alternative definition for the event contractor A 0wins in-volves the use of order statistics to highlight the ranking nature of the process.Given the random variables B 1,B 2,...,B n ,the order statistics B ͑1͒,B ͑2͒,...,B ͑n ͒are also random variables,defined by sorting the values ͑realizations ͒of B 1,B 2,...,B n in increasing order.Important order statistics,for example,are the low bid B ͑1͒=min ͑B i ͒and the high bid B ͑n ͒=max ͑B i ͒.The conditional event that contractor A 0wins,given that the average bid B ¯is between order statistics B ͑k ͒and B ͑k +1͒,is͕A 0wins ͉b 0,͑B ͑k ͒ϽB ¯ϽB ͑k +1͖͒͒=͕͑B ͑k ͒Ͻb 0ϽB ¯͉͒͑B ͑k ͒ϽB ¯ϽB ͑k +1͖͒͒͑12͒The use of order statistics will be particularly advantageous later on for determining the winning regions when using simulation.In the trivial case of only two bidders,A 0and A 1,the below-average-bid method is equivalent to the low-bid method.The low bid is the only one below the average and wins the project.For this reason,the following analysis assumes that contractor A 0faces two or more opponents ͑i.e.,there are three or more bid-ders ͒.The above expressions for the event contractor A 0wins can also be rewritten in terms of bid-to-cost ratios by dividing all bids by the cost estimate c 0,i.e.,by substituting the symbol B with the symbol X .For example͕A 0wins ͉x 0,c 0͖=͕പX i ϽX¯͓X ¯−x 0ϽX ¯−X i ͔പ͑x 0ϽX ¯͖͒͑13͒in which X i =B i /c 0and X ¯=͑x 0+X 1+¯+X n ͒/͑n +1͒.Similarly,the general formula for the probability that contrac-tor A 0wins can be expressed in a variety of ways,such asP ͓A 0wins ͉b 0͔=P ͓പB i ϽB¯͕B i Ͻb 0͖പ͑b 0ϽB ¯͔͒͑14͒P ͓A 0wins ͉x 0,c 0͔=P ͓പX i ϽX¯͕X i Ͻx 0͖പ͑x 0ϽX ¯͔͒͑15͒The conditional probability that contractor A 0wins,given that theaverage bid B ¯is between B ͑k ͒and B ͑k +1͒,isP ͓A 0wins ͉b 0,͑B ͑k ͒ϽB ¯ϽB ͑k +1͔͒͒=P ͓͑B ͑k ͒Ͻb 0ϽB ¯͉͒͑B ͑k ͒ϽB ¯ϽB ͑k +1͔͒͒͑16͒P ͓A 0wins ͉x 0,c 0,͑X ͑k ͒ϽX ¯ϽX ͑k +1͔͒͒=P ͓͑X ͑k ͒Ͻx 0ϽX ¯͉͒͑X ͑k ͒ϽX ¯ϽX ͑k +1͔͒͒͑17͒Case of Two OpponentsThe complexity of determining the probability that contractor A 0wins when using the below-average-bid method is illustrated clearly even in the simplest case of facing only two opponents,A 1and A 2.For A 0to win,given that it has selected a bid-to-cost ratio x 0,one of the following four compound events ͑inequalities ͒must be truex 0ϽX ¯ϽX 1ϽX 2͑18͒X 1Ͻx 0ϽX ¯ϽX 2͑19͒x 0ϽX ¯ϽX 2ϽX 1͑20͒X 2Ͻx 0ϽX ¯ϽX 1͑21͒Replacing X ¯with ͑x 0+X 1+X 2͒/3gives the equivalent fourevents͑2x 0ϽX 1+X 2͒പ͑x 0+X 2Ͻ2X 1͒പ͑X 1ϽX 2͒͑22͒͑X 1Ͻx 0͒പ͑2x 0ϽX 1+X 2͒പ͑x 0+X 1Ͻ2X 2͒͑23͒͑2x 0ϽX 1+X 2͒പ͑x 0+X 1Ͻ2X 2͒പ͑X 2ϽX 1͒͑24͒͑X 2Ͻx 0͒പ͑2x 0ϽX 1+X 2͒പ͑x 0+X 2Ͻ2X 1͒͑25͒The shaded areas in Fig.2show the values of ͑x 1,x 2͒defined byeach of the four events ͓Eqs.͑18͒–͑21͔͒or,equivalently,events ͑22͒–͑25͒.For the ease of notation,we assume that X 1and X 2are inde-pendent and identically distributed ͑IID ͒with a common prob-ability density function,f X ͑x ͒,and a common cumulative distribution function,F X ͑x ͒.We can take advantage of symmetry to integrate f X ͑x ͒over areas ͑b ͒and ͑a ͒to produce the following probability of winningP ͓A 0wins ͉x 0͔=2͵−ϱx 0f X ͑x 1͒ͫ͵2x 0−x 1+ϱf X ͑x 2͒dx 2ͬdx 1+2͵x 0+ϱf X ͑x 1͒ͫ͵x 12x 1−x 0f X ͑x 2͒dx 2ͬdx 1͑26͒=2͵−ϱx 0f X ͑x 1͓͒1−F X ͑2x 0−x 1͔͒dx 1+2͵x 0+ϱf X ͑x 1͓͒F X ͑2x 1−x 0͒−F X ͑x 1͔͒dx 1͑27͒=2F X ͑x 0͒−͓F X ͑x 0͔͒2−2͵−ϱx 0f X ͑x ͒F X ͑2x 0−x ͒dx+2͵x 0+ϱf X ͑x ͒F X ͑2x −x 0͒dx͑28͒To proceed further it is necessary to make specific assumptions210x x x x <<<201x x x x <<<120x x x x <<<102x x x x <<<1x 1x Fig.2.Winning regions against two opponentsD o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y S HE N Z H E N U N I V E R S I T Y o n 12/31/15. C o p y r i g h t A S C E .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .about f X ͑x ͒and evaluate the integrals using numerical techniques.The solution for any number of opponents n can be achieved much easier using simulation as described below.Simulation ApproachMonte Carlo simulation can be used to determine the probabilityof winning and to select the optimum bid-to-cost ratio x 0for a given distribution f X ͑x i ͒.We shall assume that the apparent bid-to-cost ratios,X 1,X 2,...,X n ,are IID,following a normal distribu-tion with mean m X and variance ␴X 2.In order to arrive at a numerical solution that is independent of m X and ␴X 2,the following standardized variables are definedm X Ј=m X −1␴X͑29͒x 0Ј=x 0−m X␴X͑30͒The main input to the simulation process consists of the number of competitors n ,the number of simulation sets s ,and the number of projects per simulation set m .The results presented here are based on n =2,4,8,s =1,000,and m =1,000.For each project j =1,...,m ,in each simulation set k =1,...,s ,we sample n standardized apparent bid-to-cost ratios x ijk Ј͑i =1,...,n ͒from the standard normal distribution N ͑0,1͒.Notice that the index i represents an opponent and the index j represents a project.Index k represents the simulation set.For each simulation set,the result can be visualized as a table with n columns ͑the number of opponents ͒and m rows ͑the number of simulated projects ͒filled with random numbers from the standard normal distribution N ͑0,1͒.Each table corresponds to a simula-tion set,k ,for a total of s tables.In the following discussion we shall concentrate first on a generic project ͑j ,k ͒,i.e.,a row in one such table of standard normal random numbers,and for simplicity we shall stop using indices j and k .Thus,for the generic project we sample n stan-dardized apparent bid-to-cost ratios x i Ј͑i =1,...,n ͒from the stan-dard normal distribution and compute their average,x ¯Јx ¯Ј=1n ͚i =1nx i Ј=1n͓͑n +1͒x ¯0Ј−x 0Ј͔͑31͒Notice that this average,x ¯Ј,is different from x ¯0Ј,the average of allstandardized bid-to-cost ratios,including the currently selected value of x 0Ј͑which is the decision variable ͒x ¯0Ј=1n +1͚i =0nx i Ј=1n +1͑x 0Ј+nx ¯Ј͒͑32͒Winning RegionsFor a given simulated project,i.e.,a sample of n standardizedapparent bid-to-cost ratios x i Ј͑i =1,...,n ͒,we seek the conditions that must be satisfied so that the decision variable x 0Јwould win the project.First,we will show that values of the decision vari-able,x 0Ј,greater than x ¯Ј,are also greater than the average of all bids x ¯0Јand thus cannot win the project and can be ignoredx ¯ЈϽx 0Ј⇒͑33͒nx ¯Ј+x 0ЈϽnx 0Ј+x 0Ј⇒͑34͒nx ¯Ј+x 0ЈϽ͑n +1͒x 0Ј⇒͑35͒x ¯0ЈϽx 0Ј͑36͒We now show that values of the decision variable,x 0Ј,greater than x ¯Јproduce an average of all bids x ¯0Јthat is also greater than x ¯Јx ¯ЈϽx 0Ј⇒͑37͒nx ¯Ј+x ¯ЈϽx 0Ј+nx ¯Ј⇒͑38͒͑n +1͒x ¯ЈϽx 0Ј+nx¯Ј⇒͑39͒x ¯ЈϽx ¯0Ј͑40͒Combining Eqs.͑36͒and ͑40͒gives the final resultx ¯ЈϽx 0Ј⇒x ¯ЈϽx ¯0ЈϽx 0Ј͑41͒Thus,values of the decision variable,x 0Ј,greater than x ¯Јare of no interest since they are greater than the average of all standardized bid-to-cost ratios,x ¯0Ј,and hence cannot win.Reversing the direc-tion of inequalities and following the same line of reasoning leads to the important resultx 0ЈϽx ¯Ј⇒x 0ЈϽx ¯0ЈϽx ¯Ј͑42͒Condition ͑42͒states that the decision variable,x 0Ј,must be lessthan x ¯Ј,so that it may also be less than x ¯0Ј.This is a necessary but not a sufficient condition for x 0Јto win the project.To find the sufficient conditions for x 0Јto win,it is expedient to sort the standardized bid-to-cost ratios x i Јof the n opponents inascending order to form the order statistics x ͑1͒Ј,x ͑2͒Ј,...,x ͑n ͒Ј.Clearly,the value of the average x ¯Јmust be greater than x ͑1͒Јand less than x ͑n ͒Ј.In contrast,the average x ¯0Јdepends also on theselected value for the decision variable,x 0Ј,and can have anyvalue.It can be less than x ͑1͒Јand greater than x ͑n ͒Ј.For the sake of notation,let us assume that the average of the opponents’standardized bid-cost ratios x ¯Јlies between order sta-tistics x ͑h ͒Јand x ͑h +1͒Јx ͑h ͒ЈϽx ¯ЈϽx ͑h +1͒Ј͑43͒It is clear that any selected value for the decision variable,x 0Ј,between x ͑h ͒Јand x ¯Јis certain to win the project because from Eq.͑42͒we havex ͑h ͒ЈϽx 0ЈϽx ¯Ј⇒͑44͒x ͑h ͒ЈϽx 0ЈϽx ¯0ЈϽx ¯Ј͑45͒Thus,the decision variable,x 0Ј,is closest to and less than theaverage and wins over x ͑h ͒Јwhich is the next value closest to and less than x ¯0Ј.Most people,when encountering the below-average-bid method for the first time,consider the situation described by Eq.͑44͒as the only way x 0Јcan win the project.A little reflection,however,indicates that this is far from true as shown below.Let us now assume that the decision variable,x 0Ј,lies betweenorder statistics x ͑r ͒Јand x ͑r +1͒Јwhere x ͑r ͒ЈϽx ͑h ͒Ј͑i.e.,r Ͻh ͒.For x 0Јtowin,it is sufficient thatx ͑r ͒ЈϽx 0ЈϽx ¯0ЈϽx ͑r +1͒Ј͑46͒For the rightmost inequality to be true we haveD o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y S HE N Z H E N U N I V E R S I T Y o n 12/31/15. C o p y r i g h t A S C E .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .。