THE USE OF COMPUTERS FOR STATISTICAL APPLICATION IN CONSTRUCTION MANAGEMENT

INTRODUCTION

The use of mathematical modeling and statistical applications in Construction Management is a relatively new management tool when viewed in terms of the history of construction management and construction in general. These techniques have their roots in military operations research techniques which were first developed during World War II by British mathematicians and scientists. The encouraging results achieved by the British and United States military management teams during the war attracted the attention of post-war industry management personnel who were seeking to solve complex functional specialization problems. The first widely accepted mathematical technique was linear programming and the introduction of the simplex method for solving a system of linear equations, introduced by George Dantzig in 1947, was instrumental in the fields advancement. The progress of the field of operations research parallels to a large degree the development of the digital computer as a system which could store large amounts of data, quickly and accurately retrieve data and most importantly perform computations at incredible speeds. The development of the micro-computer, with data based system software, more reliable retrieval systems, larger user memory and priced at a rate the average contractor can afford, has opened up a new area of decision making tools for the construction industry here to-fore only accessible to large industry and governmental agencies, who could afford the computers necessary to perform these tasks. The purpose of this paper is to explore some of the more readily adaptable operations research and statistical techniques which could be implemented by the construction industry, Three general areas are are targeted for consideration, they are:

All three of these areas are presented to undergraduate construction management majors at the University of Nebraska and the topics will be presented along with applicable software for implementation.

BIDDING THEORY

Bidding for work in the Construction Industry is a fact of life. While most contractors would prefer to do negotiated work, the vast majority of contractors, at least in the eastern Nebraska, find that approximately 85-95% of their yearly volume of work will come from jobs that were won in a competitive bid situation. This would imply that the bidding process is an important issue to the present day contractor.

A recent informal survey of 10 successful, established contractors in the Omaha-Lincoln (eastern Nebraska) area revealed the fact that none of the contractors were currently using any mathematical model or statistical analysis in a bidding situation. This is not surprising. When asked why this was the case, responses varied from:

If mathematical models are going to be used in bid-­ding analysis by contractors, these concerns have to be addressed. Change will ultimatel_y come from within, most likely brought about through the efforts of new graduates. To this end, educating New construction managers in the use of  mathematical modeling is of supreme importance, however, the old excuses must be addressed. Let's examine each of the 5 reasons given in their order of presentation:

One of the course objectives for the Computational Analysis and Methods course at the University of Nebraska is to examine current literature on Bidding Models, and to instruct the students in the rudiments of statistical analysis and mathematical modeling, then apply the mathematical model to a hypothetical bid situation using a computer generated approach.

There are 2 main mathematical models of competitive bidding strategies which have been developed over the past 30 years. The first model was presented by Lawrence Friedman (Friedman, 1955). This model utilizes a set of B/E (opponent bid/contractor estimate) ratios, which have been developed over a period of time when bidding against the same competitor. From this set of data a probability of winning when bidding against the competitor at a given markup can be determined. The maximum-expected profit can be determined from a graph which plots the product of Markup and probability of winning vs markup. Friedman's mathematical model comes into play when bidding against more than one bidder (a multiple bidding situation). Friedman states that the probability of winning will be the product of all the probabilities of winning which would be generated from a single competitor situation. Thus, if _your probability of beating competitor A was Pa at a certain markup, and your probability of beating competitor B was Pb at that same markup, then your probability of beating both competitor A and competitor B at the same time would be Pa x Pb. This concept Friedman exdented to any number of bidders. Thus the formula:

This is precisely the mathematical model gaming theory presents for success based upon mutually exclusive events occurring at the same time, and no proof is necessary (i.e. a head coming up on a two-sided coin and a 6 on a die happening at the same time would be 1/2 x 1/6 = 1/12).

A second model introduced by Marvin Gates (Gates, 1967) follows Friedman's model in that a set of B/E ratios for each competitor is formulated, this set of ratios is grouped and a cumulative fre­quency curve is developed (ogive or sigmoid) for ease of extracting the probability of winning against each individual competitor. The mathematical expectancy curve is also developed according to Friedman's model. The main difference between the two theories lies in the mathematical model used to determine the probability of winning against multiple bidders. Gates does not agree that the conceptual bidding situation is correctly represented as a set of mutually exclusive events but is more closely approximated by the drawing of a ball from an urn which contains many different colored balls (the number of balls of each color is representative of the probability of winning the given competitor has with respect to each of the other competitors at a given markup). This formula is given by Gates as:

An informal argument establishing the validity of this equation as being representative of the mathematical model it proports to represent is presented in the Appendix.

Subsequent authors such as Morin and Clough (1969), Benjamin (1972), Wade and Harris (1976), and Carr and Sandahl (1978), to name a few (a more complete listing appears in the list of References) have incorporated parts of one or both of these models into their models. The purpose of this discussion is simply to point out the fact that the real difference between these models is the question of which mathematical model more closely approximates the real world situation. Which assumption is correct is ultimately an empirical matter that we can and should test. This is precisely the type of modeling procedure which we try to impress upon the student. The author's experience with a number of contractors and designers who have used one or both of the models favors Gates' interpretation. To this end, a computer program based upon Gates' model was developed and implemented into the curriculum. This program takes raw data (input as bids and estimates), sorts and groups the data into classes, calculates the statistical information (mean, mode, median, standard deviation etc), constructs is used or not and should not 'prevent a person from adopting a bid model to assist in the bid generating procedure.

To summarize, the purpose of presenting this bid model to young future contractors is to dispell ,the reasons previously presented by various con­tractors for not incorporating a statistical approach to the bid process and also to acquaint the student with some of the steps required when implementing a mathematical model into a actual situation, keeping in mind that empirical evidence gathered as a result of utilization of a model may well change the character of the model.

REGRESSION ANALYSIS

The one statistical proceedural method which all contractors surveyed stated they had used was curve fitting and regression analysis. Both curve fitting and regression analysis lend themselves quite readily to computer application. A menu driven program was developed for student use which determines the least squares best fit curve for the following models:

the s-curve (ogive) for each single bidder and then constructs the mathematical expectancy curve and interpolates for the peak, thus determining the optimum markup for maximum profit. The program also allows for the use of the data bank to determine the same set of statistical answers when involved in a multiple bidder situation. Instruction is also given which would allows the student to devise his/her own program utilizing an electronic spreadsheet such as Visicalc, Lotus 1-2-3, Multiplan, etc.

Before implementation of any bidding model takes place the student/contractor must be aware of pro­blems which frequently are encountered when trying to adapt a bidding model to the real world situa­tion, some of these are listed below:

The 5 problem areas mentioned, which may pose a problem to implementation of a bidding model, are problems which should be addressed whether a mode'The output generated includes S the coefficients of the specified formular (0 <, r < 1), and the standard deviation of the y and x values. Projections using the regression formula(s) can then be made. Input includes the desired confidence level and the value of the independent variable and the output is the range of the dependent ariable as a projected value. The students are also required to perform some of the calculations on an electronic spreadsheet. This program is useful for application in cost projections on long-term duration jobs (this is a requirement on some government contracts), equipment utilization projections, and work projections based upon past experience. As with all statistical models, this data can provide the manager with a number or range of expected values which is superior to the typical "gut feeling" approach so typical of our industry.

A by-product of the study of regression analysis is the technique of solving systems of equations using matrix manipulation, in particular a Gaussian elimination method. This technique leads to the presentation of the final regression technique-multiple regression analysis. That is, a measurement of the association between several independent variables associated with a single dependent variable. The proceedure is similar to that for simple correlation with the exception that other variables are added to the regression equation. Symbolically:

TRANSPORTATION MODEL

The third topic which uses a mathematical model in solving a physical problem is the use of linear programming to solve a problem involving the allo­cation of limited resources to minimize cost. This problem is typically known as the Transportation Model (Assignment model). This model, in its basic form, seeks to determine a transportation plan which allocates a single commodity to a number of destinations using a number of sources. The basic assumption of the model is that there is a direct proportionality between transportation costs and units transported for a given route. The transportation model is basically a linear program that can be solved by the regular Simplex method and as such lends itself to a computer solution. However, the special structure of the model allows the development of a solution procedure called the transportation technique that is computationally more efficient. The beauty of the linear programm­ing approach to problem solving does not lie in the fact that a single best solution _an be found but in the sensitivity analysis (what if game`)

which can be performed on the set of equations or inequalities of the model. The optimum solution may not be the most feasible solution and the ,decision making process is greatly enhanced through utilization of this procedure.

BIBLIOGRAPHY

APPENDIX

The following argument is presented not as a formal proof, but as justification for the use of the Gates' formula to represent the probability of choosing a certain colored ball from an urn containing many balls of various colors.

Assume:

Pa is the probability of choosing a white ball from a combination of white and blue balls, Pb is the probability of choosing a white ball from a combination of white and red balls, and Pc is the probability of choosing a white ball from a combination of white and green balls. (e.g. Pa = .6, Pb = .7 and Pc = .4). The problem is to deter­mine how many white balls, red balls, blue balls and green balls to put into the urn to satisfy the given conditions. To determine the number of white balls, the numerators of all probability ratios must be the same. Then:

Thus, the number of white balls can be represented by Pa x Pb x Pc, then the number of blue balls would be Pb x Pc - Pa,x Pb x Pc, the number of red balls would be Pa x Pc - Pa x Pb x Pc, and the number of green balls would be Pa x Pb - Pa x Pb x Pc. We then have the probability of choosing a white ball from an urn containing the white, red, blue and green .ails being