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Using Spreadsheets and VBA for Teaching Civil EngineeringConcepts

Essam ZaneldinDepartment of Civil and Environmental Engineering, United Arab Emirates University

P.O. Box 17555, Al Ain, United Arab Emirates

and

Bilal El-ArissDepartment of Civil and Environmental Engineering, United Arab Emirates University

P.O. Box 17555, Al Ain, United Arab Emirates

1. ABSTRACT

Spreadsheets are becoming increasingly popular insolving engineering related problems. Among the strongfeatures of spreadsheets are their instinctive cell-basedstructure and easy to use capabilities. Excel, forexample, is a powerful spreadsheet with VBA robust

programming capabilities that can be a powerful tool forteaching civil engineering concepts. Spreadsheets cando basic calculations such as cost estimates, scheduleand cost control, and markup estimation, as well asstructural calculations of reactions, stresses, strains,deflections, and slopes. Spreadsheets can solvecomplex problems, create charts and graphs, andgenerate useful reports. This paper highlights the use ofExcel spreadsheet and VBA in teaching civilengineering concepts and creating useful applications.The focus is on concepts related to constructionmanagement and structural engineering ranging from asimple cost estimating problem to advancedapplications like the simulation using PERT and theanalysis of structural members. Several spreadsheet

were developed for time-cost tradeoff analysis, optimummarkup estimation, simulating activities with uncertaindurations, scheduling repetitive projects, schedule andcost control, and optimization of constructionoperations, and structural calculations of reactions,internal forces, stresses, strains, deflections, andslopes. Seven illustrative examples are presented todemonstrate the use of spreadsheets as a powerful toolfor teaching civil engineering concepts.

2. INTRODUCTION

Spreadsheets are among the earliest softwareinnovations that had a profound effect on the

widespread use of personal computers. Spreadsheetsmade their first appearance for personal computers in1979 in the form of VisiCalc, an application designed tohelp with accounting tasks [1; 2]. Since that time, thediversity of applications of the spreadsheet program isevidenced by its continual reappearance in scholarlyjournals. Spreadsheets are, therefore, among theearliest software innovations that had a profound effecton the widespread use of personal computers. Amongthe strong features of spreadsheets are their intuitivecell-based structure and the simple interface that is

easy to use, even for first time users [3; 4]. Underneaththe structure and the interface is a host of powerful andversatile features that can be utilized in teaching fromdata entry and manipulation to a large number offunctions, charts, and word processing capabilities.Spreadsheets can also be used to analyze multiplesolutions efficiently and accurately, and to produce

graphs that convey the solution to the readers [5]. Thisis in addition to its visual basic for applications (VBA)powerful programming capabilities. Newer spreadsheetversions have also added many productivity features forInternet connectivity, workgroup sharing, powerfulprogrammability options, and a number of add-inprograms. With their wide use, spreadsheets have beenused as tools for developing computer models that canbe used as a tool for teaching construction managementconcepts, for which ease of use, versatility, andproductivity are the main issues. Hegazy and Ersahin[3] used Excel and VBA to develop an informationsystem for subcontractors and small/medium-sizedcontractors. Their developed spreadsheet storesresource data for labor, equipment, crews, material,subcontractors, and alternative methods of constructionfor various tasks. In addition, a separate worksheet isdesigned for each project to be used for estimating andcontrol purposes. Mixon [5] used spreadsheets todevelop a statistics package for students to conductsimple ordinary least squares estimation.

Ickert and Huston [6] developed a spreadsheet that canbe used

to analyze multiple solutions for engineering

problems efficiently and accurately, and to produce

graphs that convey the solution to the end users. Thiriez[7] developed several spreadsheets as educational toolsfor students. One example is to use the drawingcapabilities of Excel to represent decision trees and

where window switching and macros allow the educatorto animate his presentation. Another example presentedby Thiriez [7] is the use of Excel in dynamicprogramming, deterministic or stochastic, where Excelfunctions facilitate the development of specializedmodels. From primary to tertiary levels, the spreadsheetis gradually increasing in its importance as a tool forteaching and learning. Robson et al. [8] presented aworkbook for estimation using Excel by providing built-indata management features and a set of menusdesigned to lead students through the data collection

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and estimation process. On the other hand, a number oftextbooks, like Levine et al. [9] are built around the useof spreadsheets for statistical analysis.

In this paper, Microsoft Excel 2007 [10] is used fordeveloping spreadsheets that can be used for teachingcivil engineering concepts. Two areas in civil

engineering were selected namely constructionmanagement and structural engineering. Spreadsheetapplications related to these two areas were developedincluding construction data management, time-costtradeoff (TCT) analysis, optimum markup estimation,simulating and scheduling construction activities withuncertain durations, scheduling linear and repetitiveprojects, schedule and cost control, and optimization ofconstruction operations, as well as the structuralcalculations of reactions, internal forces, stresses,strains, deflections, and slopes. First, the use ofspreadsheets as a tool for teaching constructionmanagement and structural engineering concepts ingeneral is discussed. Seven illustrative examples arethen presented to demonstrate how spreadsheets can

be used as a powerful tool for teaching civil engineeringconcepts.

3. SPREADSHEET AS A POWERFUL TEACHINGTOOL

Among the strong features of spreadsheets are theirintuitive cell-based structure and the simple interfacethat easy to use even for first time users. Underneaththe structure and the interface is a host of powerful andversatile features, from data entry and manipulation to alarge number of functions, charts, and word processingcapabilities. Newer spreadsheet versions have alsoadded many productivity features for Internet

connectivity, workgroup sharing, programmabilityoptions, and a number of add-in programs. With theirwide use, spreadsheets have been used as tools fordeveloping computer models in domains such asconstruction, for which ease of use, versatility, andproductivity are the main issues. In this paper, aspreadsheet program, Microsoft Excel 2007, is used todevelop applications to solve civil engineering problems.Several powerful and infrequently used Excel featuresare available and can be used to develop practical andpowerful models for teaching civil engineering concepts.These features include data lists, data menu options,data filtering, referencing and searching lists, basicspreadsheet functions such as Vlookup, and pivottable reports. In Excel, a data list is a simple structure of

columns and rows that contain data. Entering data intoa large spreadsheet list may become extremely tediousand prone to error. The Data-forms menu option inExcel provides a simple way for editing or deletingexisting records in a list or adding new records. Sortingthe data helps bring similar records together for visualinspection or other purposes such as preparing reportsand charts. Using the Data, Sort menu option, the listsdata can be arranged in an order that is chosen bysorting the records. Filtering the data is also a usefulway to view a subset of the records that compose a list.

To filter a list is to extract records from it, based oncriteria set by the user. Referencing and searching thelist is another important part of the data managementprocess. In realistic systems, where several lists of dataare available, a link needs to be established amongthem (similar to the relational database concept). Onesimple and important spreadsheet function Vlookup

can be used to link separate lists of information bymaking a reference to where the original data are.Accordingly, it is possible to determine the a cost of acertain project item, for example, by using a Vlookupfunction to search the original list and determine itsassociated rate per hour value. The Match, Index,and Offset are other functions that provide furthercontrol over the data in a list. Reporting is anotheressential requirement for obtaining summary data. InExcel, the pivot table wizard provides an automatedreport generator.

In addition to the features and functions of Excel, itsprogramming capabilities, called visual basic forapplications (VBA) are very powerful and can be used

to solve complex civil engineering problems. Excel canalso be used in linear programming, where the solver isused, and where the spreadsheet's graphingcapabilities are used both to represent the feasible setand the objective function, and to interactively animatethe objective function or constraint movements. Excelcan also be used in simulation, where stochasticsimulation may be facilitated through the use of an add-in, and a decision support system may thus be builtfrom scratch.

4. USING SPREADSHEETS IN TEACHING CIVILENGINEERING CONCEPTS

The basic paradigm of an array of rows-and-columns inExcel spreadsheets with automatic update and displayof results has been extended with libraries ofmathematical and statistical functions, versatilegraphing and charting facilities, powerful add-ins suchas Microsoft Excels Solver, attractive and highlyfunctional graphical user interfaces, and the ability towrite custom code in languages such as MicrosoftsVisual Basic for Applications. Spreadsheets with thesepowerful features and programming capabilities can beused as an effective tool for use in civil engineeringapplications. Examples include construction datamanagement, cost estimation, TCT analysis, optimummarkup estimation, simulating and schedulingconstruction activities with uncertain durations,

scheduling linear and repetitive projects, schedule andcost control, optimization of construction operations,design of bolted steel connections, and analysis anddesign of different steel members such as beams,columns, base plates, etc. Since the core of anyinformation system is the storage of the data andinformation for management operations, databasecapabilities in spreadsheet programs can be used as aneffective tool to develop complex data managementsystem for construction information. Few basic, butinfrequently used, spreadsheet features need to be

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known and can be used to develop practical andpowerful models, which can be used for teaching civilengineering concepts. In this study, severalspreadsheets were developed to be used for teachingcivil engineering concepts. In the following subsections,example applications are presented in two areas in civilengineering: 1) construction management, and 2)

structural engineering. These two areas were selectedas they represent the areas of specialty of the paperauthors.

4.1 Applications in Construction Management

Four examples are presented in this paper todemonstrate the use of spreadsheets in teachingconstruction management concepts. The first exampleillustrates the use of Excel for the management ofconstruction information. In Excel, a data list is a simplestructure of columns and rows that contain data.Entering data into a large spreadsheet list may becomeextremely tedious and prone to error. The Data-formsmenu option in Excel provides a simple way for editingor deleting existing records in a list or adding newrecords. This option can be used to develop aconstruction data management system to manageinformation related to construction equipment, labor,material, cost, methods of construction, and bills ofquantities. Sorting the data helps bring similar recordstogether for visual inspection or other purposes such aspreparing reports and charts. Using the Data-Sortmenu option, the lists data can be arranged in an orderthat is chosen by sorting the records. Filtering the datais also a useful way to view a subset of the records thatcompose a list. To filter a list is to extract records fromit, based on criteria set by the user. Referencing the listis another important part of the data managementprocess. Also, Excel has several important and powerful

functions. For Example, one simple and importantspreadsheet function Vlookup can be used to linkseparate lists of information by making a reference towhere the original data are. For example, consider thesituation when a new list is used for estimatingpurposes (Figure 1) and this list refers to the Code ofthe labor being used. Accordingly, it is possible todetermine the cost by using the Vlookup function tosearch the original labor list and determine itsassociated Rate/hr value (Figure 1).

Figure 1: Using the Vlookup Function.

If the resource code specified in cell A4 of the estimateis changed (e.g., L5 is used), the costs will be adjustedautomatically in cells C4 and D4 of the estimate. Inaddition to the Vlookup function, the Match, Index,and Offset functions provide further control over thedata in a list. The help system of Excel can be used to

obtain information about the syntax and use of theseimportant functions. Reporting is another essentialrequirement for obtaining summary data on resourcesand operations. In Excel, the pivot table wizard providesan automated report generator. Pivot tables can beused to arrange projects information in the neededformat, summarize long lists in a compact format, findrelationships within lists that are hidden by all details,display data in the form of averages, percentages,summations, etc.

The second example illustrates the use of probabilisticmodels to analyze the behavior of a contractorscompetitors bidding for a job in order to optimize themarkup used in bidding for a job. The probability

approach has historically been the most populartechnique for the construction of bidding models. Mostbidding models based on probability theory have beenderived from the work of Friedman [11]. Friedmanclaimed that, in a tender, it was possible to model eachbidder's behavior as a function of the estimated cost bymeans of a probability distribution. The generalapproach assumes that there are a number of bidderscompeting regularly against each other in the samemarket place and, given a sufficient number ofopportunities to tender against known competitors, anyone player can collect sufficient information to model therelationship between its own markup on future projectsagainst the probability of submitting the lowest tender.This approach also assumes that the basic prime cost is

similar for all competitors, and by comparing thecompetitors' tender figures for past projects with its ownestimate of prime cost, a contractor can thereforedevelop a probability distribution for each of itscompetitors showing the likelihood of winning futuretenders with different markups. Using Friedmansmodel, the probability of winning against a range ofcontractors P(winall) is the product of the chances ofbeating them each individually, and the maximumexpected profit (EP) can be calculated as follows:Expected profit (EP) = Profit x Pwinall

Where: P(winall) = Pwin1x Pwin2x x Pwinnand n: is the number of competitors.

Gates [12], on the other hand, suggested that bycomparing its own bid to the winning bid a contractorcould calculate the markup which would have beenneeded in order to win the contract. Again this impliesthat cost is the most important criterion:

P(winall) =

Vlookup function in Cell C4:

= vlookup (A4, Labour!A1:C6, 3, 0)

Search List Column Exact

value number match

1[(1-P(Win1))/ P(Win1)]+...+ [(1-P(Winn))/P(Winn)]+1

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To illustrate this concept to, the two approaches weremodeled using Excel and a spreadsheet was developedto calculate the optimum markups using both Friedmanand Gates approaches. When a contractor bids for anew job, it is assumed that he/she knows othercompetitors bidding for the same job. To estimate theoptimum markup for the contractor, the user will only

enter the number of previous bids for each competitorbidding for the job, the competitors bid values for thesebids, and the contractors estimated cost. Excel thencounts the number of bids for each competitor and the

bid to cost ratio for each bid and for and eachcompetitor. It also calculates the mean and standarddeviation of the bid/cost for previous bids of eachcompetitor. The mean is calculated based on thesummation of the bid price to bid cost ratios for allavailable bids for each competitor divided by thenumber of available bids. Excel will also calculate the

probability to win each competitor separately and theprobability to win all competitors using Friedman andGates models. Figure 2, shows an example of biddingagainst four competitors (A, B, C, and D).

Figure 2: Optimum Markup Estimation Sheet.

In this example, the maximum expected profit (max. EP)is found for both Friedman and Gates models using theMax function. As shown in Figure 2, the maximumexpected profit using Friedmans model is found to be$2,394.34 (cell I82) and using Gates model is$23,365.63 (cell J82). Excel will then find the markupcorresponding to the maximum expected profit for bothFriedman and Gates models using the Vlookupfunction. For example, the Vlookup function used to

calculate the markup corresponding to the maximumexpected profit for Friedmans model (Cell I83) =Vlookup(I82, I15:K80, 3, 0). For the example in hand,the optimum markups using Friedman and Gatesmodels were found to be 3.20% and 7.10%,respectively.

The third example illustrates the use of spreadsheets forscheduling construction activities with uncertain

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durations using the Program Evaluation and ReviewTechnique (PERT). Using this technique, each activityhas three durations; the optimistic time (a), thepessimistic time (b), and the most likely time (m). PERTuses a weighted average of the three times to find theoverall project duration. This average time is called theexpected time (te), which is equal to (a + 4m + b)/6. To

determine the probability of a project to be completedearlier or later than expected, the variance (v) of eachactivity along the critical path is calculated as follows:v = (b - a)

2/36. Since the duration of each activity is

uncertain; the time of occurrence of each activity is alsosubject to uncertainty. The measure of uncertainty ofthe final event in a PERT diagram is the standard

deviation of the expected time (TE). The TEfor the lastevent is the square root of the sum of the variance of allactivities along the critical path. Also the expected timeof the last event in the project is denoted as T E. Todetermine the probability of completing a project earlieror later than expected, the deviation (z) needs to becalculated; where z = (TS TE)/TE and TS is the

scheduled time to finish the project.

As shown in Figure 3, a spreadsheet was developed fora simple five-activity example to illustrate the PERTconcept using MonteCarlo simulation and severalduration iterations were generated.

Figure 3: An Example of Scheduling Construction Activities with Uncertain Durations.

For the five-activity project shown in Figure 3, the userneeds to enter two times only (the optimistic time aand the pessimistic time b) for each activity. Activitiesstandard deviations () and variances (v=

2) will then

be calculated automatically. Several iterations are thengenerated by selecting random numbers between theoptimistic and pessimistic times using the Excel rand()function. For the purpose of this example, five iterationsare generated. Each path in the network is thenidentified and, following the logic of the network, thecritical path is automatically calculated by identifying the

longest path in the project. The expected time of the lastevent in the project (TE) is calculated and the standarddeviation (TE) for critical activities in each path for eachof the five iterations is also calculated. Excel thencalculates the average expected time (TE) and theaverage standard deviation (TE). For a scheduled time(TS) = 48 days; for example, the probability (Prob.) tocomplete the project in this duration using iteration 3 iscalculated using the normdist function and found to be67.08%. For example, using iteration 3, the number ofdays that corresponds to completing the project with a

Calculating the

probability & risk for

a given project

duration

Calculating the

projectsduration for

a given probability

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probability of 67.08% is calculated using the norminvfunction and found to be 48 days.

The last example is a VBA application. In this example,an Excel spreadsheet was developed and a VBA macrocode was written for time-cost trade-of analysis. Thesheet provides information about project activities with

their logical relationships and their correspondingdurations, resource requirements, methods ofconstruction, and costs associated with each method ofconstruction. The VBA macro was developed to find theoptimum method of construction for all project activities.The basic premise of the model is to allocate optionalconstruction methods for each activity, varying fromcheap and lengthy to expensive and short. Using themodel, users will be able to select different constructionmethods and find the corresponding total project cost.The proposed model, as such, is usable not only at theplanning stage but also during construction. The VBAmacro then automatically generates a Microsoft Projectfile for the project using the data of the spreadsheet.The code was written to generate a Microsoft Project file

for the project example in hand. The code copiesactivities data such as activities names, durations, andstart times, construction methods, and activities actualcosts from the spreadsheet and then transfers the datato a newly generated Microsoft Project file for theproject. A complete schedule for the project using theselected data is finally automatically generated.

4.2 Applications in Structural Engineering

Three examples are presented in this paper todemonstrate the use of spreadsheets in teachingstructural engineering concepts. The first exampleillustrates the use of Excel to enhance the concept andefficiency of moment distribution method whenspreadsheet is used. A continuous beam subjected touniform distributed load was used (Figure 4) to illustratethe concept and to show that spreadsheets reduce thenumber of random arithmetic errors and time in doingthe tedious moment-distribution computations.

Figure 4: Moment Distribution Calculation Sheet.

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Moment distribution method is a classical method thathas been included in every civil engineering program. Itis one of the first numerical methods for structuralanalysis. The procedure of the method for prismaticmembers can be summarized as follows: 1) computethe relative stiffness k = I/L for each member that iseither connected or supported at both ends. This

excludes overhang members; 2) compute thedistribution factor (DF) for each member connected to ajoint or support; 3) compute fixed end moments (FEM)for members with intermediate loads; 4) compute endmoments for overhanging members; 5) pre-determine atolerance limit; 6) balance the moments at each jointand distribute the balancing moment according to DF ofeach member; 7) carry over the balanced moment tothe adjacent end of each member; 8) repeat steps 6 and7 until the unbalanced moments are within the tolerancelimit; and 9) sum the moments in each column for eachmember end.

The second example illustrates the construction of theshear force and bending moment diagrams for any

structural member using a spreadsheet. Figure 5 showsthe shear force and bending moment diagrams for thecontinuous beam used in the first example (the momentdistribution example). Beams are long and slenderstructural elements that support transverse as well asaxial loads. Figure 5 shows the bending moment andshear force values at different span distances whileFigure 6 shows the bending moment and shear forcediagrams.

Figure 5: Bending Moment and Shear Force Values.

The values shown in Figure 5 are calculated usingExcel functions and the bending moment and shearforce diagrams shown in Figure 6 are automaticallygenerated using Excel charts. These diagrams areimportant in structural engineering as they show notonly the variation of the internal forces along the lengthof the structural member but also the most critical

values that the structural engineer needs to design therelevant structural member. They are the most populartechniques to show the changes in the internal forces ofa structural member along its length. These diagramsare obtained using the relationships between theexternal loads on the member and the internal forcesgenerated in the member and the free body diagrams(FBDs) to the left or right of sections that cut themember transversely and show all the forces on thoseFBDs. Then equations of equilibrium are applied toensure equilibrium is satisfied. The shearing forces andthe ending moments obtained are basically the internalstress distribution resultants. Typical examples ofbeams include bridges, roof structural elements, floorbeams, machine elements such as levers, cranks,

manipulator arms, etc.

Figure 6: Bending Moment and Shear ForceDiagrams.

The third example illustrates the use of Excel incalculating the internal stresses and deflections in abeam. Having calculated the design bending momentand shear force for the beam above, all that remains tobe done now is to assess the size and strength of thebeam. The process involved in a selection depends onwhether the construction material behaves elastically or

Bending Moment Diagram (BMD)

Shear Force Diagram (SFD)

Span (ft)

Span (ft)

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inlastically. For the demonstration in this paper, elasticbehavior has been presented. The calculation of theinternal stresses, whose resultants are the shearingforces and bending moments produced by the externalloads, and the deflections in a beam of know sectionand geometry uses the section modulus (S) and thecross section area (A) and its moment of inertia (I). As

shown in Figure 7, the spreadsheet reduces thearithmetic errors of computing the S, A, and I, thefrustrations, and the time doing the tedious stresscalculations. The procedure for computing stress brieflyinvolves: 1) computing the maximum moment anddetermine the required section modulus (S) using the

flexural formula, S = M/fb; 2) selecting the mosteconomical (lightest) structural steel shape using theAISC Allowable Stress Design Method; 3) checking thebeam for shear by comparing the actual shearing stressand the allowable shearing stress; and 4) computing themaximum deflection due to the design load andcompare it with the allowable deflection. The internal

stresses and deflections in beams are calculatedautomatically using the spreadsheet shown in Figure 7below.

Figure 7: Section Properties and Stress Calculations.

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5. CONCLUSIONS

Spreadsheets are becoming increasingly popular foruse as a powerful tool and are considered among theearliest software innovations that had a profound effecton the widespread use of personal computers. Amongthe strong features of spreadsheets are their intuitivecell-based structure and the simple interface that is

easy in education. This is in addition to its powerfulprogramming capabilities. Underneath the structure andthe interface is a host of powerful and versatile featuresthat can be utilized in teaching, from data entry andmanipulation to a large number of powerful functions. Inthis paper, several spreadsheet applications aredeveloped for use in teaching civil engineeringconcepts. For the purpose of illustration, seven of thesespreadsheet applications are then presented. Theexamples demonstrate the simple and powerful featuresof spreadsheets and its capability in solving complexconstruction management and structural engineeringproblems. These examples can be used by civilengineering students as templates for solving similarproblems or for developing models to solve otherproblems. In addition, the developed spreadsheets, assuch, represent a transparent methodology that allowsfor quick what-if analysis in cost estimation, optimummarkup estimation, risk analysis, and structuralmembers optimization. Finally, the availability of Exceland its VBA programming capabilities makes it apowerful tool in developing applications for teaching civilengineering concepts and concepts related to otherengineering disciplines.

6. REFERENCES

[1] J. Baker and S. Sugden, Spreadsheets inEducationthe First 25 Years,SIE journal, Vol. 1, No.

1, 2003, pp. 18-43.

[2] E. Zaneldin and S. Ashur, Using Spreadsheets as aTool in Teaching Construction Management Conceptsand Applications, The 2008 American Society ofEngineering Education (ASEE) Annual Conference,June 22-25, 2008, Pittsburgh, PA, USA.

[3] T. Hegazy and T. Ersahin, Simplified SpreadsheetSolutions I: Subcontractor Information System,Journalof Construction Engineering and Management,ASCE, Vol. 127, No. 6, 2001, pp. 461468.

[4] T. Hegazy and T. Ersahin, Simplified SpreadsheetSolutions II: Overall Schedule Optimization,Journal ofConstruction Engineering and Management, ASCE,Vol. 127, No. 6, 2001, pp. 469475.

[5] J. Mixon, GRETL: An Econometrics Package forTeaching and Research, Managerial FinanceJournal, Vol. 36, No. 1, 2010, pp. 71-81.

[6] R. Ickert and A. Hutson, Hydraulic Modeling ofTransmission Systems Using Spreadsheets,The 2005Pipeline Division Specialty Conference, August 2124, 2005, Houston, Texas, USA.

[7] H. Thiriez, Improved OR Education Through theUse of Spreadsheet Models, European Journal of

Operational Research, Vol. 135, No. 3, 2001, pp. 461-476.

[8] G.S. Robson, Y.B. Shin, and J.W. Mixon, EstimatingAccounting and Finance Models with Microsoft Excel,Managerial Finance Journal, Vol. 35, No. 5, 2009, pp.451-460.

[9] D.M. Levine, D. Stephan, T.C. Krehbiel, and M.L.Berenson, Statistics for Managers using MicrosoftExcel, 5th ed., Prentice-Hall, Upper Saddle River, NJ,2008.

[10] Microsoft Excel Reference Guide 2007. MicrosoftCorp., Redmond, Wash.

[11] L. Friedman, Competitive-Bidding Strategy,Operations Research, Vol. 4, 1956, pp. 104112.

[12] M. Gates, Bidding Strategies and Probabilities,Journal of Construction Division, ASCE, Vol. 93, No.1, 1967, pp. 75107.