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THE USE OF PROBABILITY AND EXPECTED UTILITY THEORY FOR RISK MANAGEMENT IN CONSTRUCTION

Jac de Jong

Department of Construction Science

Texas A&M University

College Station, Texas

Introduction

The majority of industries are governed by a few large capital intensive organizations, controlling most of the industry's assets. Although the heavy construction industry follows more or less the same pattern, the building industry, on the contrary, consists of a conglomeration of diverse specialty companies, brought together when a new project is initiated. The industry is fragmented to the extent that a general contractor may require some 15 to 20 odd sub­contractors. The diversity of these organizations of sub­contractors makes the coordination for planning and sched­uling by the general contractor extremely complex when a project is newly acquired.

Besides these complexities many uncertainties exist in terms of the reliability of the multitude of the different bid prices obtained from many sub-contractors, the securing of a skilled labor force, as all of them are very labor intensive, and the allocation of labor and equipment resources with the acquisition, delivery and installation of building materials.

The market itself is mostly unpredictable, strongly com­petitive and fluctuates with the economic conditions 1o­cally, regionally and nationally, not to mention the often extreme seasonal variations in the different regions.

Most projects are obtained on a competitive bid basis, usually one project at a time, and independent of all other projects, either in progress or still waiting to be tendered. Hirings and layoffs are therefore quite often the order of the day. This requires that the industry must have a built-in flexibility in order to cope with these fluctuations in the labor force to increase or decrease personnel quickly and efficiently, while still maintaining a skilled skeleton labor force to maintain stability.

Inflation and interest rates are unpredictable for projects of long duration. To obtain insurance, bid and performance bonds, for incorporation into the bid price, can sometimes be difficult and laborious, specifically under the present economic constraints.

Labor unrest always looms on the horizon. The industry is therefore probably one of the most volatile, competitive and, judging by the bankruptcy rate, least profitable busi­ness in existence.

These difficulties raise questions such as what projects to bid on to maintain a balanced operation for the company, who are the competitors and how many are there, how to establish the profit margins and what does it depend on in order to be successful in obtaining a project. In answering these questions, decisions need to be made on a responsible and rational basis in order to minimize risk and maximize profits. In the majority of cases decisions are not made in a consistent and organized manner, in particular the finalizing of the bid proposal. Therefore, the concepts of both responsibility and rationality are being lost in the process and for that reason are further explored.

What is the meaning of "responsible" and "rational" in this context? According to the "Larousse Illustrated International Encyclopedia and Dictionary", 1972, pg. 753, it means "fit to be placed in control". The "Webster's New 20th Century Dictionary", 2nd Edition, pg. 1543, defines "responsible" as "the ability to distinguish between right and wrong and to think and act rationally, and hence accountable for one's behavior".

"Rational" according to Webster's New World Dictionary of the American Language", 2nd College Edition, pg. 1179, implies the ability to reason logically as by drawing conclusions by inferences and often connotes the absence of emotionalism. This definition uses the word "reason", given as something to think out systematically by drawing of inferences or conclusions from known or assumed facts. (Webster's, pg.1183).

Considering these definitions acting "responsible" is behavioral related and requires the distinction between right and wrong and may well mean different things in different cultures. It definitely involves acting on one's own and being held accountable.

"Rational" is used in this paper as it is in the Sciences, a systematic progression or a step by step reasoned and consistent development of coming to conclusions. And, as in the Sciences, it inherently adheres to a body of conventions that establishes rational and consistent thinking. What these conventions are and the calculus of using them will be the subject matter of this paper.

Decision Analysis

Decision analysis incorporates both uncertainty and risk in the decision making process. To deal with "uncertainties" use is made of probability theory. It will be assumed that a basic knowledge of set theory and probability theory have been acquired. The axioms of probability, with the formules for calculating both conditional probability and the Bayes' theorem, are given as they relate directly to the material.

Risk is incorporated through the use of Utility theory, which relates the company's value system to the risk involved. This value system depends on the "state" of the organization itself, on the value the owner as an individual or the corporation as a group places on everything it owns and owes, including its labor force. Expanding these resources depends on the risk the company is willing to take, as the taking of risks is directly related to the return on investment.

In financial decisions, as for example those made in the bid proposal of a construction company, uncertainty is assessed using probability theory. Three different approaches for establishing the neccesary probabilities exist presently, namely:

The objective approach,

The subjective approach,

The classical approach.

The objective approach is based on the relative frequency of events in repeated experiments. Experiment here, may be interpreted in its broadest sense, as this approach has been taken by the contractor in recording the number and prices for each project.

The subjective approach gives the probability value as a measure of belief that one has in the knowledge that exists of the world at that specific point in time. This may be called experience or intuition.

The classical approach is based on the fact that events have an equally likely chance of occurrence, as for example taking a card out of a well-shuffled deck of cards. These approaches are made operative by using a calculus of probability based on axioms which will be spelled out further on. (Goldberg, 1986)

Making decisions is the process of selecting one action from an array of alternative courses of action with each alternative course of action having a specific outcome. Each outcome is evaluated in terms of its consequences for either loss or gain. The "expected outcome" is calculated by multiplying the outcome, or payoff, with its probabilities for both winning and losing and then summing the results. An individual can determine the maximum expected value of the different outcomes that satisfies the proposed goal, in this case, maximizing the return on investment. It is assumed that an individual will choose the outcome, that maximizes the expected value, as each outcome is based on the premise of anticipating the predicted future for each alternative.

The scale used for measuring these consequences or uncertainties is given by a number between zero and one, which indicates the value of the judgements of the relative likelyhood for these predictions. These numbers must satisfy the axioms of probability, given as follows:

This concept can be expanded upon in a more generalized form. The sample space S is divided into 3 mutually exclusive events. Such a division is called partitioning of the sample space. An event B intersects with all of the 3 events, A,, Az, A;, as shown in the Venn Diagram below, figure 1, then the event B may be calculated as the union of 3 mutually exclusive events.

In terms of probabilities and using the addition rule, the probability of B may be stated as follows:

This process may be described as the adjustment of the probabilities of Ai when the probabilities of B are known. One way of establishing the probability of B is by performing experiments. The "experiment" used here is the recording of previous bid proposals submitted by the competitors. When the probability p(Ai) does incorporate this experimental data, then it is called the "a posterior" probability of Ai, otherwise it is the "a priori" probability. Graphically this may be depicted as follows, figure 2:

It may also be mentioned that independence of events A and B is obtained when p(A I B) = p(A) and p(B I A) = p(B), which means that the probability of A is not altered by the probability of B, when p(B) is known. This concept is important as independence of events make the use of the "multiplication" rule possible.

Utility theory, on the other hand, attempts to describe man's behavior in which decisions involving risk are made. It consists of a preference system, called utility, that ranks the preference among different alternatives. Its scale is totally arbitrary. These preferences adhere strictly to axioms in order to be consistent and rational. These axioms are briefly stated as follows:

Axiom#1: Comparability: alternative A1 preferred to A2 or A2 to A1 or otherwise both are equally preferred.

Axiom #2: Transitivity: A 1 preferred to A2 and A2 to A3 then A 1 preferred to A3.

Axiom #3: Continuity: A I preferred to A2 and A2 to A3 then there exists a probability p for which a sure amount for A2 is equivalent to the gamble p times A1 and (1-p) times A3.

Axiom #4: Compound lottery: may be substituted for a simple lottery, which is obtained by multiplying the compound lottery by its probabilities.

Axiom #5: Monotonicity: comparing two outcomes for two alternatives, in which the outcome with the highest probability is preferred.

These are given here without proof and the reader is referred to the literature for further examination. (Lifson, 1982; Watson, 1987; Kleindorfer, 1993).

Based on these axioms the theory establishes a functional relationship, called a "utility" function, between the value of an outcome and its rank on a scale of preferences.

The function is obtained by making "equivalence propositions" between a lottery and a fixed sum of money in the following manner, figure 3, or the fixed sum of money is given and the probabilities need to be determined.

By varying the amounts of money for the lottery and the fixed sum, multiple points of the function can be established and the curve plotted.

The dollar value of each alternative is then scaled against the utility function and its utility value, measured in utiles, determined. By calculating the "expected" utility value for each outcome the contractor will prefer the outcome with the highest expected utility. (Lifson, 1982; Watson, 1987; Kleindorfer, 1993).

Two specific problems are considered, although this methodology is applicable to many problems in the construction industry.

First: the problem of establishing what project to bid on out of a choice of three.

Second: what profit margin to consider regardless of what project is chosen.

Assume that three projects are simultaneously up for tender and that the following hypothetical information is obtained in regard to the three projects.

For project 1 the contractor has a 75'/. chance of winning S 150,000.00 and a 25%. chance of losing 530,000.00.

For project 2 there is a 80% chance of winning S I 10,000.00 and a 20%. chance of losing $20,000.00.

For project 3 a 90%. chance of winning $80,000.00 and a 10% chance of losing $10,000.00.

The dollar amounts for winning or losing are the estimated quantities established by the contractor.

The given chances or probabilities are "subjective" probabilities, in other words a gut feeling the contractor has about the chances of winning or losing.

To be able to chose the best alternative, in this case one of the three projects, use is made of the concept for optimization of the expected monetary value (EMV). The EMV is calculated in the usual manner by adding the amounts one stands to win and lose multiplied by respective probabilities.

The calculations are performed in table 1:

When strictly going by the concept of the EMV, the contractor should choose the alternative, which maximizes the return on investment. Project I provides the maximum expected profit of $105,000.00 and thusly should be the one to select. It provides also the greatest risk opportunity in that there exist a 25% chance of losing $30,000.00.

The contractor is risk averse and wants to postpone the decision on which project to bid. An utility function needs to be established to incorporate the contractor's value system and attitude towards risk. In order to be able to construct the curve for the utility function, two approaches are followed to obtain the "equivalence propositions".

the "certainty equivalence" and

the "probability equivalence" approach.

First the range of the amount of profit and loss is calculated and the extreme dollar values determined. The maximim amount to be gained is $150,000.00, the maximum amount to be lost is -$30,000.00. These amounts are plotted on the X-axis and a scale determined.

The utility value for the $150,000.00 is set to one utile, and for the -$30,000.00 to zero utiles. A scale is established between 0 and 1, using equal intervals, on the Y-axis.

For the "certainty equivalent" approach the following hypothetical proposition is made. There exists a 50-50 chance of making $150,000.00 or losing $30,000.00 against getting for sure an amount of x dollars. This amount needs to be determined by the individual or organization involved, in this case the contractor. Establishing this amount indicates that the individual is indifferent between taking the sure financial award or the lottery with the 50-50 chance of winning or losing. (Kleindorfer, 1993) Graphically the proposition is depicted in figure 4 as follows:

Suppose that the sure amount is $5,000.00. This corresponds to the.5 mark on the utility scale, and establishes one point on the utility curve. This point signifies that the contractor is indifferent between either getting the $5,000.00 for sure or the 50-50 chance of winning the lottery of $150,000.00 or losing $30,000.00.

This process is continued by now considering $5,000.00 with a utility value of .5 and the $150,000.00 with an utility value of 1. If for example, the contractor is willing to settle for $40,000.00 for sure or the 50-50 chance for the lottery of making $150,000.00 or making $5,000.00, then the utility value of the $40,000.00 is halfway between.5 and 1 on the utility scale and is equal to .75.

The same approach is followed for the proposition of the 50-50 chance for the lottery of winning $5,000.00 with a utility value of.5 or losing $30,000.00, with a utility value of 0. It is assumed that the contractor is indifferent if the amount is -$15,000.00, which is equivalent to an utility value of .25. Five points of the utility curve are now obtained and the function can be established.

In order to generate a utility function use can be made of the PROC NLIN command in the SAS language. This requires the establishment of 15 to 20 points before the curve can be generated and the equation formulated. This seems hardly practical as each point is obtained by making a judgement between a lottery and a fixed amount of money. Besides, the preferences on the utility scale have arbitrary values in the sense that one alternative is preferred over another qualitatively but not quantitatively. How much more one alternative is preferred over the other is not specified.

The accurate "e - functions" found in the literature, (Lifson, 1982), required the determination of at least 15 individual points before such an equation could have been formulated.

Another method of generating a curve is by using the "cubic splines interpolation" method in the Mathcad software. This method fits a curve to a set of points so that the first and second derivatives at the given points of the curve are continuous, creating a series of cubic equations between the points. No actual single equation is established.

Utility values are determined as the intersection with the ordinate for the profit amounts on the X-axis and the utility curve. The utility function for the contractor is drawn by hand as sketched in figure 5.

By scaling the amounts of profits/loss to be made for each project on the utility function, the utility values for each profit/loss margin is measured. These values are (table 2):

The expected utility value is determined for each of the three projects and the maximum utility value observed, table 3.

Table 3 provides the answer to the question of what project to bid on. The project to be considered for tendering is the one with the maximum utility value of .954, which is project 3. The above utility figures give a good indication of the preference the contractor has for bidding projects involving risk.

The "probability equivalent" approach follows the same pattern with this difference that in the proposition the amount for sure is given, but the probabilities for the lottery need to be determined.

This approach is graphically depicted in figure 6:

The probability p is so determined and the process follows through in the same way as before with the "certainty equivalent" approach. This is another way for determining the same utility curve.

The second problem of what profit margin to incorporate in the bid price will now be addressed.

In the example to be used, the cost estimate of the contractor for previous bids is taken as the base-line for establishing the markups of the bids for the competitors.

Of the 20 bids recorded for each competitor, assuming that the three competitors all bid on the same projects, the percentage over/ under the cost estimate is given in table 4.

It is also assumed that the 20 jobs listed are all more or less of the same size and price range, so that the effect of job size does not influence the profit margins. Each bidder is also equally desirous of obtaining work, although the percentages of the mark-ups should reflect this trend.

The low bidder for the three competitors for each project is denoted by having brackets around the percentage number as indicated in table 4. For example, competitor 1 is the low bidder for project 1, while for project 3 competitor 3 is the low bidder. Table 4 gives the low bidder for each project.

The contractor's bid proposal, for each mark-up level ranging from 0 to 6 %, is compared with the lowest bid of the competitor for that project, as given in table 4. The opportunity for being successful in obtaining the project is so determined.

At the cost, or the 0% level of the mark‑up, the contractor's bid is compared with the lowest bid of the three competitors for each project. It may be observed that competitor 1 is still the low bidder with the bid proposal being 1 % below the cost estimate of the contractor. This is indicated in table 5 by WO denoting a low bid or a win for competitor 1 at the 0% level of the cost estimate for the contractor. This process is repeated for the 1 % level of the cost estimate with the results shown in table 5. The W 1 in the C2 column for project 7 indicates that if the contractor adds a mark-up of 1 % competitor 2 is still the low bidder. Continuing in the same manner, for all levels of the mark-ups, table 5 is completed.

Table 6 summarizes the results of table S, giving at each mark-up level for the contractor the number of projects underbid by each of the competitors. Thus at the 2% level of mark-up competitor 1 underbid the contractor on 1 more project, namely project 2, and the contractor has thus been underbid an accumulative total of 4 projects, projects: 1 (WO), 2 (W2), 13 (W 1), and 20 (W 1). Table 6 has so been tabulated, and the cumulative probabilities included.

The numbers in the cumulative row in table 6 are the probabilities that the contractor will lose the bid against three competitors. The probabilities for the contractor to win against three competitors is obtained by deducting the cumulative probabilities from 1 and given in table 7.

The probabilities in table 7 are obtained from the data given in table 4 by calculating the actual relative frequencies, thus representing the best possible estimate obtainable for these probabilities.

Other methodologies can be found in the literature. One of the methodologies makes use of the multiplication rule and establishes for each individual competitor the probability of winning a project and multiplying these individual probabilities in order to obtain the winning probability for the contractor. (Park, 1966). These may be calculated by following the same procedure as before and compare the contractor's bid proposal with the one for each of the competitors on an individual basis.

The data for that comparison is available from table 4 and the results for each competitor, are given in table 8. The cumulative probabilities for the contractor to win bids against the individual competitor C1, C2, and C3 are summerized in table 9 by deducting the cumulative probabilities in table 8 from 1.

By using the multiplication rule, the probabilities for winning against each individual competitor may be multiplied together in order to obtain the probabilities for the contractor to win a bid against three competitors.

At the 0% mark-up level the contractor's probability of winning would then be .95 . .95 . .95 = .857. At 1 %: .689, 2%:.48, 3%:.232, 4%:.149, 5%:.023, 6%:.005.

These values are compared with the ones in table 7, which were obtained by using the relative frequency approach, and are summarized in table 10.

Although these figures are remarkably similar, the multiplication rule requires that events are independent. The contractor's bid price and those of the three competitors are not independent events. On the contrary they are mutually exclusive and exhaustive, which means that the occurrence of one event precludes the occurrence of the other and that only one of the events can occur. When events are mutually exclusive they cannot be independent. (Hays, 1973). Independence exist only if the probability of one event is not influenced by the other.

Assume that G denotes the contractor, then for independence to hold p(G I C 1) = p(G), for all Ci's, i = 1, 2, 3. This indicates that if the probability of C 1 is known the probability of G would stay the same and the multiplication rule would be valid, which is not the case.

This controversy brought about the suggestion that the Bayes' theorem would be a better approach towards establishing the probabilities for the contractor to win against the three competitors. (Lifson, 1982). In order to use the Bayes' theorem the events need to be dependent and mutually exclusive. Although not individually independent, pair-wise independence may well exist. (Goldberg, 1986), in which case this scenario may be described as a pair-wise comparison among the pairs (G - C1), (G - C2) and (G - C3). For the multiplication rule to be valid independence must then exist among the pairs (C 1 - C2), (C 1 - C3) and (C2 - C3).

Using the Bayes' theorem this pair-wise comparison can be shown diagrammatically in a probability tree as given in figure 7.

The .95 probabilities in figure 7 are obtained from table 9 at the level of 0% mark-up.

The "a priori" probability for each competitor is 1 /3, as each has an equally likely chance of occurring, if the assumptions hold.

The "a posterior" probability is then calculated by using the Bayes' formula as follows: p(C1 /W 1) = p(C1) , p(W1 / C1) / (p(C1) . p(W1 / C1) + p(C2). p(W2 / C2) + p(C3). p(W3 / C3)]. (Goldberg, 1986).

The probability of winning for the contractor, at the level of 0% mark-up, against the three competitors is the summation of W1 +W2 +W3 = 1/3 . (.95) + 1/3 . (.95) + 1/3 . (.95) = .95. This amounts to nothing more then averaging the individual probabilities for the three competitors. This is accomplished for each level of mark-up and the results tabulated in table 1.

The same table gives the comparison with the probabilities in table 10. Theoretically, pair-wise independence has not been dealt with abundantly in the literature, and its validity here is questioned when the results are compared with the ones given in table 11.

For purposes of father analysis the probabilities, obtained from the methodology of the actual determined relative frequencies, will be used.

Assume that the contractor's cost estimate is $2,500,000.00 and that the loss of investment, if bid is lost, is $20,000.00.

To be able to chose an alternative, in this case one of the projects, again use is made of the concept for optimization of the expected monetary value (EMS, in the manner as used before.

In tabular form the EMV's are summerized in table 12:

As can be seen from the calculations, the maximum amount of expected profit is $15,000.00.

Adhering again to the concept of the EMV, the contractor should chose the alternative, which maximizes the return on investment, which is the one with the $15,000.00 profit at the 2% level of the mark-up. The probability of winning the bid is .5 or 50%. However the contractor is anxious to employ the available resources and may decide not to persue this particular option. Instead the contractor will be looking for the alternative that best serves the objectives for keeping his work force productive and taking the appropriate kind of risk.

Use is therefore made of the utility function as previously established in figure 5.

By scaling the amounts of profits to be made for each level of mark-up on the utility function, the utility values for each profit margin is measured.

The utility value for losing the bid is .19 utiles for the $20,000.00, as measured from the utility curve. Summerizing, these values are given in table 13.

Given the figures in table 13 the expected utility values can be determined by multiplying the probabilities of winning and losing with the respective utility values for the profits as indicated. Table 14 gives these calculations. The alternative with the highest utility value, which is the alternative at the 1 % level of the mark-up, is chosen for the bid proposal. As can be seen this alternative has a probability .650 or 65% of being successful and deviates from the choice made under the EMV calculations, which was the third alternative at the 2% level of the mark-up.

In conclusion it may be stated that the methodology for establishing probabilities by using relative frequencies in order for the contractor to win certain projects appears to be the most feasible, but also the most cumbersome.

Epilogue

In order to improve decision making a clear rational approach, consistent with the principles and axioms spelled out herein, is necessary. Uncertainty is involved in every decision problem to be faced. Probability as a measure of uncertainty maybe based on both judgmental factors or historical data, as is the case herein.

The subject of judgement has its inherent ramifications. But so has the relative frequency approach. Who is to say that the pattern from historical data will follow through into the next problem situation, indicating that regardless of the results obtained judgement still needs to be exercised.

Using utility theory for risk assessment also has to overcome the hurdles of judgemental factors by substituting certainty for a lottery. These judgements are no doubt subject to human error. However these will improve as experience is gained and this approach is used as a means to provide a consistent and rational methodology for gaining meaningful experience.

References

Goldberg, Samuel. 1986. Probability. An Introduction. Dover Publication, Inc., Prentice Hall, Inc., New Jersey.

Hays, Wiliam L. 1973. Statistics for the Social Sciences, Holt, Rinehart and Winston, Inc., New York, N. Y.

Meindorfer, Paul R, Kunreuther, Howard C., Schoemaker, Paul J. H. 1993. Decision Sciences. Cambridge University Press, New York, N. Y.

Lifson, Melvin W., Shaifer, Jr.,Edward F., 1982. Decision and Risk Analysis for Construction Management. John Wiley & Sons, New York, N. Y.

Miller, Irwin, John E. Freund and Richard A.Johnson.1990. Probability and Statistics for Engineers. Prentice Hall, Inc., New Jersey.

Papoulis, Athanasios. 1990. Probability and Statistics. Prentice Hall, Inc., New Jersey.

Park, William R 1966. The Strategy of Contracting for Profit. Prentice Hall, Inc., New Jersey.

Watson, Stephen R. and Dennis M. Buede. 1987. Decision Synthesis. Cambridge University Press, Cambridge, U. K.