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It may take up to minutes before you receive it. The file will be sent to your Kindle account. Disclaimer:A readable copy. All pages are intact, and the cover is intact. Pages can include considerable notes-in pen or highlighter-but the notes cannot obscure the text. As part of RPRA, we will review elementary probability theory and statistics. The various features of this textbook make it possible for engineering students to become well versed in the 'machinery' of probability theory. They also make the book a useful resource for self-study by practicing engineers and researchers who need a more thorough grasp of particular topics.

For version Summer , a number of new examples, updates and references have been inserted throughout the text. Last edited by Tygokasa. Probability concepts in engineering planning and design Alfredo H-S Ang.

Contributions Tang, Wilson H. Share this book. Legal advocacy for older persons at risk. Therese Neumann, a stigmatist of our day. It appears that the risk-aversiveness of a decision maker increases the value of additional information.

The mathematical forms of utility functions commonly used to model such risk-aversive behavior would include the following. A normalized logarithmic function may be as follows Quadratic Type 2. Figare 2JO Logarithmic utility function. The coefficient measures the negatiye curvature of the utility.

For the exponential utility function, the risk- aversiveness can be shown, using Eq. Observe that the coefficient of risk-aversion does not vary with the attribute in the case of the exponential utility function ; whereas in the case of the logarithmic. In such cases, the preference behavior of the decision maker is referred to as "risk-affinitive.

It is believed that this preference -behavior is ordinarily not realistic. The correct choice of the form of the utility function, however, may not be very crucial, especially- if the expected utility values are not sensitive to the form of the function.

In order to examine the sensitivity ofthe expected utility associated with a given action to the above three forms of utility functions, consider the simple case in which the possible outcomes from an action can be described by the value of a random variable X.

Using the second-order approximation to evaluate the expected utility see Chapter 4, Vol. Var X. Hence, the exact form of. As indicated in Eq. In practice, the first two statistical moments could be all the information that may be available for a random variable; hence, Eq. Probability could be estimated froni the relative frequency of observed events. The third approach is commonly referred to as the Bayesian approach in which the judgmental information of the decisio.

Examples of these have been presented in Chapter 8 of Vol. In both the second and the third approaches, the accuracy of the probability estimation will rely on how weil. In the foilowing sections, some of the methods for obtaining these. A simple branch in a decision tree. If the decision maker is- indifferent between the lottery in Fig. With the proposed design A, he; expects that the structure will have only 0. Let x 0 take on an initial value of 0.

If design B is preferred by the engineer, x 0 will be increased slightly, say 0. Oilfereniial Settlement in. This procedure continues until an indifference condition is achieved between the two designs.

A common method for this purpose is the fractile method. The decision maker would first estimate a value such that the random variable will be equally likely to exceed. As a consistency check, the decision maker may ask himself, "Is it equally likely that the random variable would be within the range x 0.

The fractile values. As an illustration, Fig. Curve B. Based on this. For a more detailed discussion and illustrations o the fractile method, readers may refer to Raiffa and Schlaifer The fractile method could be very time-consuming and mentally stren. Quite often the available jnfotmation -is limited. In this case. For instance, a uniform distd. Based on this subjectively assumed distribution. Table 6. The main objectives of the proposed waste disposal scheme might include the minimization ofenvironmental pollution.

These multiattribute functions would evaluate the joirit utility value of several measures of effectiveness, toward fulfilling the various objeCtives. Churchman, Ackoft and Arnoff suggested a simplified model. A relatively simple and direct method is provided for evaluating a diverse set of objectives in terms of Weighted relative utility values, and selecting the alternative that best balances the competing objectives.

Stimson applied it to public health planning. The model may also have implications in engineering decision problems; the basic concepts of this niodei are presented in the next section.

The relative merits of importance of the various objectives may be different. The information required for the weighted objective decision analysis can be summarized as shown in Table Table 2. First, the n objectives a,re listed in. In this step, a preference statement is also solicited from the decision maker With respect to combinations of objectives. Starting with the most important objective, it is assigned an arbitrary weight, for example, Next, the numerical weights are assigned to each of the other objectives.

The set of initial weight assignment is then cross-checked for consistency with the preference statements established in the ordinal ranking. For instance, in a water supply planning project, if 0 1 denotes the objective of reducing the demand for water and 0 2 denotes the objective of increasing the water supply, then the importance of achieving both objectives would.

As long as extreme care is exercised to insure that the objectives are mutually exclusive with respect to the contributions to the overall utility, linearity should yield useful results. For example, the problem previously mentioned could be avoided if the two objectives are combined and denoted as no water shortage.

In the event that inconsistency in tb,e preference statement iii found, the decision maker is asked to revise his preference statement or his assigm:ilent of relative weights or both, until all inconsisten' ies are eliminated. Finally, the overall relatiye utility o each alternative is computed as. The optimal alternative is the one.

Observe that the absolute numerical value of ui is not important; the relative value of ul is sufficient for the selection of the optimal alternative. This includes. The concept of a regional. The region is primarily a poor agricultural area with a minimum number of industrial concerns.

Improving job opportunities for loc:al resi- dents is desired. Because of the natural beauty and character of this mountain region, the planners felt that the conservation of the natural state is imperative. Weighing of Objecrives. Although each plan could be implemented. The plan fully exploits the recreational potential of the area. This plan would retain much of the natural character of the area while providing. Efficiency l de. Agriculture 0 0. Utility The overall relative utility for each alternative is then computed according to Eq.

A comparison of the overall relative utilities for the various alternatives indicates that alternative al, providing limited n:c:reation, scores the maximum numerical utility values. Observe that although al is the optimal alternative, it fails to provide the maximum possible contribution to any single objective.

Tnis is a case study of the water supply problem in' San Angelo. Moreover, there is a general lack of ,systematic resource planning and mlinagement. Jid la. In order to estimate the relative weights for the. Using ordina and cardinal rankings, each respondent reported his or her assigned weights for the fiye objectives together with a justification for such weight. These individual assessments were summarized and then circu- lated among all the experts involved.

Each individual could revise his assessments at this point. The updated TCS! From 30 years of observed runoff data for the Concho Rivers. At the year the projected overall water demand is estimated to be 22, acre-ft. Based on data compiled by Freese. Observe that the demand. Hence, the probability of meeting the damand 'in the year is simply the probability that the net annual surface water available exceeds its demand for each alternative.

These probabilities are computed to be 0. To evaluate each alternative with respect to the improvement of water quality, interviews with local political. Based on the limited information and reports by Freer. Nichols and Endress , , ratings for each alter:native with respect to the water quality were established, respectively, as 0. These values indicate that the quality of surface water is judged inferior. The rating is equivalent, in a relative sense, to the probability of each alternative achieving the objective Qf good water quality.

Similarly, ratings lor additional recreational benefits are established as 0. The costs of the altemati ves in most cases were estimated by informed officials. In order to convert this to numerical ratings, the alternative with the lowest annual cost is assigned a score of 1. Thus, a nting of 0. The overall relative ut!

For example, the overall relative utility for a 1 lS. The decision maker can directly assess the weights and probabilities as appropriate. In other words, he has complete know- ledge of the entire process. A more rigorous approach to decision analysis with multiple objectives is presented in Sectiori 2. Further- more, the limitations of the linearity assumption should be recognized. Alternatively, if the riatural.

It is obvious that each of these attributes will reqUire its respective units of measurement. For example dollars, minutes, and CNR are standard measures of cost, time, and noise level, respectively. The corresponding utility function and the associated probability density function will, therefore, be multidimensional. Determining these joint utility and density functions requires the evaluation of the conditional utility and probability functions.

Some of these assumptions are as-follows.. Utility Independence The relative utillty of X ; remams the same-regardleS. The procedure is described in the following. Then, from Eq; 2. U Indifferent lotteries. It can be shown that the joint function will be a sum of various products of marginal utility functions. However, if the contractor rents modem equipment, the time of completion will range between 2 and 4 weeks Fig. Applying a linear transiormation.

Similarly, thc. Waves are the major loading on the platform. The distribution of the annual maximum wave height H is lognormal with a median of 20m and c. Mean c. H exceeds 28m and will collapse when H exceeds 32m. The respective consequences associated with lhe damage and collapse states of the two proposed designs are sliilllDarized in Table E E, where the consequences X and N for each path arc described in terms of the resPective means and standard deviations l, u.

The probability of collapse C or damage D or intact I associatc:O with each proposed design can be computed from the given statistical distribution on wave heights. The Alternative Sites Because of severe environmental constraints, there arc only two sites suitable for a large, international airport in the Mexico City metropolitan area. One is the Texcoco site of the existing airport.

The site is close to the city; however, it has been surrounded on three sides by mixed residential and cotnmercial developments, and thus, further expansion is severely limited. This site is also on top of a thick alluvium that has caused considerable differential settlement problems in the past The second site is in Zumpango.

The Zumpango site is farther from the city, has more room for expansion, and is on higher and firmer ground. To evaluate the potential consequences of the di:lferent alternatives, it is necessary to study the impacts of the airport sites on these interest groups.

Six major objectives were identified as appropriate, as follows: 1. Minimize total construction and maintenance costs. X z '"' Hourly capacity in terms 'or the number of aircraft operations per hour.

For simplicity, however, the X,'s were assumed. In most cues, dat2 were available to validate and. C:onfirm the judg- mental probability measures.

Hence, the derived densitY functions represent the judgment of a group of people familiar with the problem and the impacts of the different alternatives. As an example, Fig. The six marginal utility functions u.

As an example of these results, the utility function for mean. The procedure summarized in Eqs. Change of Effectiveness with Time Since a duration of 30 years is involved, the measure of effectiveness associated with each objective for each alternative may not be constant throughout this period, or the achievement of the objective may occur at different points in time. For all but one of the other measures of effectiveness. Specifically, the average access time, the expected number of casualties per accident in the year period, the total number of people displaced..

Results The expected utilities fer each of the feasible alternatives were obtained using a computer program developed for this purpose. The -main river runs from the Carpathian mountains and after flowing through the flat Hungarian Plain, joins the Danube River in Yugoslavia.

System 1 A multipurpose canal-reservoir system to transfer water between the Tisz. In this system, the natural supply of water is available only 4 to 5 months per year. It would usc all of the natural supply of this part of the Tisza River basin but not all of the available storage capacity. System A reservoir system will be developed on the ftatland part of the. Only a limited volume of 5. For this system to function properly, extensive and long-term international cooperation would be necessary.

System 5 a Involves!! The system would be a conjunctive groundwater system utilizing the Tisza River water for both supply and groundwater exchange:The maximum yield of tbe aquifer is a limiting factor. Objectives and Attributes The basic purpt. The plans call for signifi. Lure, industry, and hydroelectric power generation. For example, it was found that.

It was found thanhe ccr:tainty equivalent oi 98 did not change when only the levels of the other attributes x 2 to. The analyses of the other attributes established that they too were utility-independent. Table Ea summarizes. The uncertainties l! S50ciated with these attribute levels are assumed to be negligible: By substituting these attnoute levels into the joint utility function, the utility corresponding to each of the five proposed systems are determin:rl as:.

This occUrred at. The respec P2: 1. For example, the travel time between A and C is N S. A bus company is. The sequence of service is not imponant. Assume that the time spent at each station is negligible. If one of the roads was to be improved so trui. Repeat this if the objective is c. A service-station owner on the corner of the intersection estimates that his loss as a function of the project duration is as follows: ,. The causeway is a system of girders resting on piers.

As the span length of the girder increases, the construction cost per foot of girder decreases, whereas the cost of pier construction increases. Dis the present design system capacity; and a: is an additional safety factor that could be imposed to achieve a h1ghcr reliability.

The ainount of seepage recorded during the test period is classified into three categories: low, medium, and high. If low seepage is indicated. If medium seepage is indicated, Q1 and Q2 will be equally likely.

The total cost of the field test program is 3 units. From feasibility studies, three alternative sizes of the housing project were suggested for further study, namely: " 1: units a2 : units a3 : units in the first phase with an optional seco. Less than units 0.

His alternatives are to submit bids for both projects, or for either project, or none at all. Based on 'his experience and considering the expected. The estimated profit on j. U he fails to win any bid, the contractor will lose S; this includes the cost of employees and equipment left idle. Assume that the probabilities of winning projects A and B are. Construct the decision tree and. The spillway system may fail. With the available information, the lifetime failure probabilities for spillway systems A and B arc estimated to be 0.

In the judgment of the engineer, there is a 0. If the test indicates a preference for system A, system B may still be built but the failure probability will be increased to 0. On the other hand, if the test indicates that system B is preferred, its failure probability will be 0. Assume that decisions will be based on EMV. Because of time and manpower limitations, he cannot alford to submit bids for both projects.

The corresponding probabilities for the annual largest earthquake are 0. I:IF- Q Determine the optimal decision. Assume that the prediction models apply only to the improved road scheme, whereas the. The meeting is scheduled to begin in another 90 minutes. He judges that if he stays within the legal speed limit.

In this case, however. Suppose that the loss a. If the engineer is ticketed. Assume there is no other loss. Formulate the decision problem and determine his optimal ovcrspeed.

Both of these materials will last for 10 years with proper maintenance. The annual maintenance cost will depend on the quality of the pavement material as well as the volume of traffic. Low 0. However, the prediction from this research group is not per- fectly reliable. If the actual future traffic is high H , the group will predict high traffic H. Similarly, P L. Should such a study be performed before the selection of the pave:ncnt material?

The filter is packed with particles so that a large surface area is provided for the oxidation and purification process as the waste water is sprinkled from the top arid allowed to percolate through the particles. There is. LYSIS '. The plaooer is given the 'option whereby the quality of the workmanship may be tested by a week's trial performance. Draw the. Each of the n! Assume that't le tinie between:bre8lcdowns for each machine is T; which is-exponentially.

Assume that there are only two possible outcomes in the trial period: the equipment will or will not break- down.

There arc two modes of failure associated with each design. Let IX. The performance indicators are all statistically independent random variables as follows: IX. Suppose the initial costs of design A and design Bare, respectively, 1. Moreover, failure in modes r and 2 will incur ad- ditional costs of S5 million and S2 million, respectively. Assume that a design can be subject to both modes of failure and all costs are additive.

The survival of a specific load test will imply that IX11 will be at least 0. Determine the value of this load tcsL c In part a , if. Construct-the decision trec.

Numerical integration may be needed.. In a project. Geologic G o. The material is also suitable for business and economics students who have studied calculus. This site comply with DMCA digital copyright. We do not store files not owned by us, or without the permission of the owner. We also do not have links that lead to sites DMCA copyright infringement.