Decision under uncertainty applied to a hydrologic problem.

Item Type Thesis-Reproduction (electronic); text

Authors Dvoranchik, William Michael,1946-

Publisher The University of Arizona.

Rights Copyright © is held by the author. Digital access to this materialis made possible by the University Libraries, University of Arizona.Further transmission, reproduction or presentation (such aspublic display or performance) of protected items is prohibitedexcept with permission of the author.

Download date 30/05/2021 03:26:56

Link to Item http://hdl.handle.net/10150/191543

DECISION UNDER UNCERTAINTY

APPLIFD TO A HYDROLOGIC PROBTFM

by

William Michael Dvoranchik

A Thesis Submitted to the Faculty of the

DEPARTMENT OF SYSTEMS ENGINEERING

In Partial Fulfillment of the RequirementsFor the Degree of

MASTER OF SCIENCE

In the Graduate College

THE UNIVERSITY OF ARIZONA

1 g 7 1

STATEMENT BY AUTHOR

This thesis has been submitted in partial fulfillment of re-quirements for an advanced degree at The University of Arizona and isdeposited in the University Library to be made available to borrpgersunder rules of the Library.

Brief quotations from this thesis are allowable without specialpermission, provided that accurate acknowledgment of source is made.Requests for permission for extended quotation from or reproduction ofthis manuscript in whole or in part may be granted by the head of themajor department or the Dean of the Graduate College when in his ju.d.2- -ment the proposed use of the material is in the interests of scholar-ship. In all other instances, however, permission must be obtainedfrom the author.

S IGNED: L3 A1LayA r.)t&1,4 TD -NTUU-x/vta,L,

APPROVAL BY THESIS DIRECTOR

This thesis has been approved on the date shown below:

A) tr -

DateTi I

Lucien Ducks teinProfessor of Systems Engineering

ACEMWLEDS\i IENTS

This study could not have been what it is without the assis-

tance and understanding of many people. First, the author is indebted

to his advisor, Dr. Lucien Duckstein, whose assistance, comments, and

understanding were a major motivating force in completing this under-

taking. Second, the author is indebted to Dr. Chester C. Kisiel, who

provided the opportunity for the author to enter the field of Hydrol -

ogy. His patience, enthusiasm, and support were most helpful in con-

cluding this study. Since this study is a synthesis of the knowledge

acquired during the Master's Degree program, the author wishes to

thank the many faculty members, who contributed their time and effort

to make this program successful.

Also, the author is indebted to MY. Donald R. Davis, whose

hours of "talk" sessions enlightened and aided the author's under-

standing of the problem. The author is grateful to Professor Samuel

R. Browning for supplying a double integration computer program.

This research was supported in part by research grant B-007-

ARIZONA on the "Efficiency of data collection systems in hydrology and

water resources for prediction and control" from the Office of Water

Resources Research, United States Department of the Interior.

The author would also like to thank Miss Bonnie Jo Barthold

for typing this thesis in so short a time.

TABLE OF CONTENTS

Page

LIST OF TABLES vi

LIST OF ILLUSTRATIONS vii

ABSTRACT viii

CHAPTER

1. INTRODUCTION 1

2. PROBLal DEFINITION 4

Basic Problem 4Bayes Risk and Opportunity Loss 6Expected Opportunity Loss 7Expected Expected Opportunity Loss 8Depth-Flow Relationship 9Probability of Bridge Failure 10Decision Theoretic Approach Applied to this Problem . 12

3. PROGRAMMING METHODOLOGY 14

Search by Golden Section 14Double Integration Routine 20Error Function for Probability of Bridge Failure 22

4. SENSITIVITY AND RESULTS 25

Choices of Error Limits for Program Efficiency . • 25Results 30Sensitivity of Objective Function 30

Sample Size, or Selection of Different Years'Data 31Selection of Different Underlying Distribution . . 32Changes in the Bridge and Pier Costs 34Change of the Depth-Discharge Relationship . . 35

5. CONCLUSIONS 36

APPENDIX A: LISTING OF VARIABLES 37

iv

11-

TABLE OF CONTENTS--Continued

Page

APPENDIX B: GENERATION OF NORMAL CHI-SQUARE WITHCONCOMITANT UNCERTAINTY DISTRIBUTION 39

APPENDIX C: LISTING OF COMPUTER PROGRAM AND RESULTS 44

REFERENCES 68

LIST OF TABLES

Table Page

1. Comparison of Integration Limits 26

2. Computer Sensitivity Comparison 29

3. Comparison of Different Sample Sizes or DifferentSelection of Years 32

vi

LIST OF ILLUSTRATIONS

Figure Page

1. Division of line segment into 'golden ratios' ' 15

2. Jumping if f(pt(2)) < f(pt(1)) 17

3. Jumping if f(pt(2)) > f(pt(1)) 18

4. Deletion of extraneous point 19

5. Cumulative density functions for different underlyingdistributions for the streamflow data 33

vii

ABSTRACT

This thesis provides a basis to evaluate the worth of addi-

tional information. A decision theoretic framework is employed to

help delineate the components of the decision process.

Design modifications of a specific problem are the variables

that cause a change in the evaluation of the worth of additional infor-

mation, but the design Chosen is based on the minimum Bayes risk when

the structure is finally built. The hydrologic problem involves the

annual peak flows for a river and their effect on the design problem.

The process presented takes a great amount of uncertainty from

the question, "Do I have enough data on which to base a good decision?"

Assignment of a dollar value to the worth of an additional data

point tells the decision maker what he can expect to gain by waiting

an extra time period for that data point based on knowledge possessed

at the present time.

vi;;

CHAPTER 1

INTRODUCTION

Throughout history man has been forced to make decisions

based on the data he has or could get. The question that continually

arises in the decision maker's mind is, "Do I get enough information

from the data I possess or should I collect more data to reduce the un-

certainty of the decision?" Herfindahl (1969) points out that infor-

mation follows economies of scale, such that the more information one

gathers, the less incremental value is gained from this additional

information. It is important to realize that if the addition of new

data cannot alter the action to be taken, then it is senseless to

accumulate new data. In other -words, the only time additional infor-

mation possesses a value is when the action to be taken can be al-

tered with the addition of new information.

Inherent in every decision process is an amount of uncertain-

ty, particularly about the true state of nature. If every parameter

in the decision process were known for sure, there would be no need

for decision makers, since the decision process would consist of

nothing more than calculating the appropriate action based on the true

states of nature.

Of course, it is evident that the true states of irture can

never be known for certain, so a framework for making an intelligent

2

decision about the problem in question is presented here, along with

a quantitative examination of a bridge and pier design problem to

show the expected losses from insufficient tnowledge and expected

gains from additional information in a practically oriented environ-

ment.

The following analysis is done in a decision theoretic frame-

work. A decision theoretic approach is presented to formalize the

thought process and provide a foundation for arriving at an intelligent

decision. It is by no means complete or exhaustive. The approach is

as follows:

1) Define the decision and possible alternatives.

2) Select an appropriate objective or utility function to

be evaluated.

a) Select the state or input variables.

b) Develop the stochastic properties of the state

variables.

c) Establish a time horizon for the decision.

d) Include possible risk aversion.

3) Select a decision based on present knowledge by evaluating

the expected value of the objective function for each al-

ternative and choose the alternative that maximizes this

expected value.

4) Analyze the uncertainty(ies).

a) Determine the expected opportunity loss due to uncer-

tainty.

3

b) Determine the expected reduction in expected oppor-

tunity loss and the total cost of gathering the infor-

mation that caused this reduction.

A time horizon and risk aversion will be omitted from this

analysis, as are the monetary considerations, such as discount factor

or return on investment; but discussion of these items and their effect

on the decision process is covered by Raiffa (1968). Other aspects of

the decision process that could be pertinent and may contribute to the

decision are discussed subjectively by Myers and Melcher (1969).

The method of solution is Bayesian in nature with a more com-

plete analysis of the theory given in Raiffa (1968) and Raiffa and

Schlaifer (1961).

It is the purpose of this thesis to demonstrate that the theo-

retic concepts about decision theory can be taken out of the texts and

classrooms and applied to a practical situation --in this case, a

bridge and pier design problem based on the annual peak flows of a

river, in which a dollar value is placed on the worth of additional

information, which is defined as the expected reduction of the Expected

Opportunity Loss, less the full cost of obtaining this infolmation.

Hopefully, this will provide the decision maker with a valuable tool

for making an intelligent decision with regard to collecting more data.

The last point, which hopefully is brought out in this the-

sis, concerns the fact that the Systems Engineering discipline enabled

the composition of this thesis, which combines varied engineering

fields and practices, demonstrating the worth of interdisciplinary stud-

4- 4- 1,ies anu Lu uranch out into other fields of research.

CHAPTER 2

PROBLEM DEFINITION

The illustrative problem concerns the worth of additional data

for the design of a bridge over the Rillito Creek, a stream near

Tucson, Arizona.

To avoid a deep analysis into bridge design, the work of

Laursen (1969), concerning bridge design and scour analysis, is ad-

. justed for use in this thesis.

Basic Problem

Laursen assumes a 500-foot bridge is to be built atop flood

dikes that rest on four piers, each containing 25 piles. The cost of

the bridge that may be lost in a flood is $150,000. The cost of sink-

ing piles is approximately $4.00 per foot or $400/ft for the entire

structure.

The problem is to detelmine how deep to put the piles in order

to minimize the cost due to sinking the piles, plus the expected cost

of replacing the bridge, caused by a loss of the bridge during a flood,

because the piles were undercut by scour. This is the variable cost of

the bridge and is defined as:

V(h, u,2) = BC • P25 + h • 400

-where h is the pile depth in feet, u and a

2 are the mean and variance

(2.1)

4

of the annual peak flows for the Rillito Creek. The computation of P25

• embodies the parameters u and 2 and will be discussed later.

The largest contribution in terms of dollar cost to the above

objective function is caused by the bridge structure. To reduce this

contribution, the probability of bridge failure must be made small,

which causes the depth of piling to be increased, but this contribution

is negligible compared to the cost of the bridge. This tradeoff is the

basis for the minimization problem.

The data needed for this analysis concerns annual peak flood

magnitudes for the Rillito Creek. Ten years of data (1951-1960) were

used for the formal analysis, making a sample size of 10. For the en-

tire data record of 46 years, a two-parameter log normal distribution

was hypothesized and a Kolmogorov -Smirnov Test did not dispute this

hypothesis.

Uncertainty enters the analysis in the fouit of lack of 'true'

or exact knowledge of the parameters of the log normal distribution

representing peak flows. If the two parameters, the mean and variance,

were known, then the exact probability of bridge failure could be cal-

culated for any depth of piling, which reduces this problem to a simple

numerical minimization. Since the parameters are not known for sure,

uncertainty is encoded into the problem by assigning probability distri-

bution functions to the parameters in the following manner.

The application of a log normal distribution for the data en-

ables the employment of normal theory on the transformed data, where it

is known from classical statistics that the distribution of the sample

mean is normal with mean u and variance 02/n, and the distribution of

6

the sans of squares is chi-square with n-1 degrees of freedom. The

probability distribution to encode the joint uncertainty is the joint

density of the normal and chi-square distributions, since the uncertain

parameters are the mean and variance of the log normal distribution. An

.appropriate transformation is made from X to ns2 / 2 in0 the chi-square

portion of this density function and this yields a normal gamma dis-

tribution as indicated by Raiffa and Schlaifer (1961), which will be

used synonymously with normal chi-square.

Bayes Risk and Opportunity Loss

The uncertainty of the state variables, u and a 2 , adds risk to

the decision process. The distribution of this joint uncertainty is

known, so minimizing the objective function over the pile depth by tak-

ing the expectation of the objective function with respect to the joint

uncertainty distribution yields the Bayes Risk of the decision process.

This is accomplished by using a 'weighted average.' Each point of the

joint uncertainty distribution has a likelihood associated with that

point. By multiplying the value of the objective function at each pos-

sible point by its likelihood at that point and summing, we get a risk

evaluation for a specific pile depth; the minimum of this process is

called the Bayes Risk and is denoted as:

R(h*, m, s2 , n-1) = Min [E 2[V(h, u, 02)]]h u, G

where h* is the value of h that minimizes the Bayes risk.

(2.2)

7

The concept of opportunity loss is introduced; it represents

a measure of the value of perfect information on the population's para-

meters. The true parameters of the population are denoted as ut and

This information would yield ht , the depth of piling that yields

the minimum variable cost,

2B(t'h u ' a

2) = MinMin [C(h, ut, a

t)]t t (2.3)

Having used h* instead of ht'

an opportunity loss has been realized,

because the optimal pile depth for the Bayes risk will differ from the

optimal pile depth if u and a 2 were known, unless the sample and popu-

lation parameters are one and the same, so V(h*, ut , a t) > V(ht , ut , a t).

2 2For ut and at , ht is the value which minimizes V(h, ut , at); hence any

value h*, that is not equal to h, yields a V(h*, u ' a2 ) that must bet t

greater than the value using ht . Thus, the opportunity loss is:

2 2 2OL(h*, ut , at) = V(h*, ut , a t) - V(ht , ut , at) (2.4)

It is the upper bound for a consulting fee in order to obtain perfect

information about next year's maximum flow.

Expected Opportunity Loss

To solve this problem, the 'weighted' averages for the opportu-

nity losses are calculated, where each opportunity loss is weighted by

the 'probability' or likelihood that it was calculated with the true

values of u and a2 . In this manner the expected opportunity loss (XOL)

8

is arrived at by taking the expectation of the opportunity losses with

respect to the normal chi-square using m, s - , and n-1 as parameters.

Thus,

XOL(h*, m, s 2 , = E 2 -fV(h*' u ' c - - V( t'h u

' o2 )] (2.5)t t t t'

where h* is the piling depth chosen on present data and ht

is the piling

depth chosen when u and e 2 are assumed known.

Expected Expected Opportunity Loss

The expected value of perfect infoimation is known, but what

is the value of one more sample -- of one more year's data? If someone

could state the exact value of next year's maximum peak flow, an XOL

could be calculated. Include this flow in the augmented data base.

Any reduction in the value of the XOL caused by the additional data point

is the value of the sample information, denoted VSI. If this VSI is

less in dollar value than the cost of obtaining the sample, it would be

valuable to wait and get next year's peak flow before building the

bridge.

Needless to say, no one can accurately state next year's maxi-

mum flow. To circumvent this problem an average of the VSI's is taken

over all possible values for next year's flow, weighting each value by

the 'probability' of obtaining that flow. This is accomplished by

taking an expectation of the expected opportunity loss of next year's

data. The expected expected opportunity loss (XXOL) is:

Ut,

at

9

_XXOL(m, s', ) = E [NOL(h*' m' s-x' n+1)]x x (2.6)

where h*, m, and s - are calculated by . including the new data point x.x x

The expectation over x is based on the normal chi-square with concomi-

tant uncertainty distribution (see Appendix B for development), which

yields:

n1/2

f(xls 2 , n-1) - (ns2)(n-1)/2 r(n/2) (2.7)

1/2 1/2 2(n+1) 7 ((114-1)Sx) n/2 r((n -1)/ 2 )

Thus, the expected value of sample information, which is our measure of

the worth of additional data is,

EVSI = XOL - XXOL (2.8)

This is a subtraction of expected opportunity losses because EVSI is

the reduction in XOL caused by the addition of a new data point to the

data base. It is the dollar worth of new information.

Based on the EVSI, a design could be made to build the bridge

to withstand a specified flood or wait for more data, in which case,

the money that may have been spent on the bridge could accumulate in-

terest or some type of return, but there would also be a loss of bene-

fits from the Bridge.

Depth-Flow Relationship

In order to solve the problem, a relationship between the depth

of scour and flow must be developed. This is done by using a power

function to the 3/5 power, as suggested by Laursen (1969). From the U.S.

Corps of Engineers Plan I (1967) the following is depicted:

10

d = K • Q3/5 (2.9)

Values for d and Q are 20 ft. and 85000 cfs., respectively, so that K

has a value of 2.2 • 10 -2. From Laursen (1969) Fig. 4, for five feet

of scour and 15 feet of flow, it is shown that

h = M • d1/2

(2.10)

and therefore, M = 2.58. Laursen (1969) suggests using a multiplying

coefficient of 0.9 for semi-circular noses on piers, so Eq. 2.10 be-

comes

h = (2.58) • d1/2 • (0.9)

and substituting in for d yields

h = (2.58) • (0.9) • (0.022 •

and simplifying,

h = .344 • Q3110

which is the empirical relationship used in this problem.

(2.11)

(2.12)

(2.13)

Probability of Bridge Failure

Another relationship needed in the problem is the probability

of the bridge failing at least once in its life span of twenty-five

years. This twenty-five year bridge life is variable and subjective,

but it is meant to account for obsolescence and depreciation. It is

assumed that a new bridge will be built after twenty-five years and

will be based on the costs at that time.

11

It is possible that during the twenty-five year life span of the

bridge a flow of enough magnitude to destroy the bridge could occur

several times, or not at all. The binomial distribution describes this

relationship as follows, where y is the number of floods in j years and

p is the probability of failure in any one year, which is independent

and constant from year to year ,

P(Y) = qT) PY (1-P) i-Y•(2 . 14 )

Since this is a probability density function, it is known that

(i ) PY (1-P) i-Y = 1 .y=0

Thus,

P(y at least once) = 1 - P(never),

and substituting the values of the problem yields

P(y at least once) = 1 - (1-025

(2.15)

(2.16)

(2.17)

This is the probability of bridge failure at least once in twenty-five

years. To determine this quantity, the probability of failure in one

year must be ascertained. This is accomplished by assuming a design

with a certain pile depth. One converts the pile depth into flow that

it protects against, and using the log normal distribution, computes the

probability of getting a flow greater than the flow protected against by

the pile depth.

12

Decision Theoretic Approach Applied to this Problem

The problem in terms of the decision theoretic approach, out-

lined in the introduction, will now be discussed to demonstrate that

adherence to a logical thought process can yield an intelligent deci-

sion.

The decision to be made is to build a bridge to a design speci-

fication based on the depth of the piles or to wait for another year's

data before building the bridge.

The objective function was previously defined as

V(h, u, cy 2) = BC P25 + h • 400

(2.18)

or, in words, the cost of constructing the bridge multiplied by the

probability of bridge failure at least once in twenty-five years, plus

the pile depth times the cost of sinking the piles one foot. This

yields an evaluation in dollar units.

The state variables for the problem consist of the pile depth

and the parameters of the distribution of the peak flows. These vari-

ables have been discussed previously.

A time horizon and risk aversion were mentioned in the intro-

duction.

The evaluation as described by Eq. 2.2 is calculated and this

yields the optimal design based on the present knowledge, that is, the

design which has the minimum expected cost associated with it.

Expected opportunity loss caused by the uncertainty of the mean

and variance of the log normal distribution is computed by Eq. 2.5.

13

Then uncertainty about next year's maximum flow is encoded into

the problem by the same computations as above for all the possible

points for next year's flow, which yields the expected expected oppor-

tunity loss. The reduction in the XOL from year to year is the dollar

value of the worth of additional information, which leaves only the

decision as to collection of more data or building the bridge now, ac-

cording to the optimal design strategy.

CHAPTER 3

PROGRA\E\IING IETIAODOLOGY

Given the preceding theoretical concepts, a CDC 6400 computer

was used to obtain numerical answers, but the program is adaptable to

other computer systems. A brief description of the processes that were

employed to achieve the desired results follow and the entire FORTRAN

computer program is presented in Appendix C.

Search by Golden Section

A search routine for the minimum of the Bayes risk and XOL was

used and it took the form of a 'Golden Section' iterative process, as

described by Wilde and Beightler (1967). This method of elimination

takes its basis from a 'golden ratio' and enables the deletion of ex-

terior points by subdividing the present interval under consideration

into preset ratios as the Pythagorean Brotherhood did many centuries

ago. This division of a line segment is shown in Fig. 1 below.

The following relationships hold:

BC _ BC - AD 1.618033989...AB - CD AC = BD - (3.1)

This search by golden section is used because it is an efficient

manner of seeking an optimum when the number of trials is unknown. The

process of determining the optimum starts by choosing a good guess for

a starting point, then jumping one unit to the right for the second

point. Since we are searching for a minimum, and not a maximum, we are

14

D

Fig. 1. Division of line segmentinto 'golden ratios'

15

16

looking for a 'desired bracket' or interval of three points, where the

function evaluated at the outer two points is greater than the evalu-

ation at the interior point. Hence, if the second point's evaluation is

less than the evaluation of the first point, the next jump is in the

same direction, but it is the previous interval times T, where T is the

golden ratio of 1.6180.... If the second point's evaluation was greater

than the evaluation of the first point, the direction of the jump is

from the first point in the opposite direction and is equal to the inter-

val size times T. Figs. 2 and 3 give a graphic illustration.

This jumping continues until the desired interval bracketing

is found. Since each jump is approximately 1.6130 times the last jump,

this bracketing procedure rapidly covers the minimum, especially with a

good first guess. Once the bracketed interval is determined, it is

subdivided as in Fig. 4 and each point is evaluated. One of the extreme

points is then deleted depending upon the evaluation of the points. The

point that is dropped is the extraneous point outside the desired bracket.

Fig. 4 gives an example.

Point A is deleted because points B, C, and D faun a desired

bracket. This procedure continues until If(B) - f(D)I<E, where 6 is a

predetermined constant, which is defined in a later section. At this

time, the midpoint of the interval is evaluated and the minimum is taken

to the value of the midpoint or the value of the interior point, which-

ever is smaller.

There are additional ways to increase the speed and accuracy of

this searching process, but this was deemed to be of sufficient accuracy

pt (1) pt C2) pt C3)

Fig. 2. Jumping if f (pt (2)) < f (pt (1))

17

f (X)

1 • T

f(x)

ls

1 • T

pt(3) Pt(1) pt(2)

Fig. 3. Jumping if f(pt(2)) > f(pt(1))

f (x)

A B C D

extraneous desired bracketing

Fig. 4. Deletion of extraneous point

19

20

and sPeed for this problem, so that other techniques were not imple-

mented.

Double Integration Routine

In much of the aforementioned theory, the term expectation was

used. To take the expectation over u and e2 entails a double integia-

tion procedure. This is accomplished by combining two one-dimensional

Simpson's rules, which is an approximate integration procedure. The

routine solves an integral of the form:

b h(x)

S f (w, x) dw dx .

a g (x)

The adaption of this routine to this specific problem is

(3.2)

where,

F(u, (5 2 ) • Nx 2 (u, a2) du do-2 (3.3)

2a = ns2 / df • x 99.9 (df),

2h = ns2 / df • x .1 (df),

geM) = m - 3 • s,

h(M) = m + 3 • s,

F(u, e 2) = V(h, u, e 2) ,

21

(u-m) 22

ns"2 ,, (df/2) -12,Nx 2 (u, 0_2 ) _ (ns L) rh,

r(dfr) 7dT/2 (27)112 el--20""

The limits for a and b are set up in this fashion, so that the computer

program is adaptable for other sample sizes ranging from 1 to 44.

These values for a and b are taken from the CRC Handbook of Ta-

bles for Probability and Statistics (1968) for the Chi-square distribution

with nine degrees of freedom, since n is equal to 10, so that the inte-

gral includes 99.8 percent of the area of this distribution. Similarly,

g(m) and h(m) are designed to include over 99.7 percent of the area of

the normal distribution. N5( 2 (u, a 2) was derived by combining the normal

and chi-square density functions and making an appropriate change in2

variables from X to , so that this function is actually a normala

gamma, as discussed by Raiffa and Schlaifer (1961). To facilitate

faster integration, the integral is broken into two segments, from the

lower limit a, to the mode of the chi-svare, (ns2)/(df-2), and from

this mode to the upper limit b. This is more efficient because the

slopes of the two segments change less rapidly than the slope of the

entire range of the chi-square distribution.

The manner in which the integration is done is similar to a

rectangular grid network, such that for each point on the horizontal

axis a one dimensional Simpson's rule is performed over the vertical

axis.

When two iterations are within 10 percent of each other, the

routine teuainates. With this error level, the noimal chi-square was

found to evaluate to over 99 percent of its combined area.

22

Similarly, a. one dimensional Simpson's rule is used to take the

expectation of next year's possible data points. This is of the form:

d2

f(s -' n) ds 2

x x 'c

(3.4)

where,

= mx -4. sx

d

= mx + 4 • sx m

n1/2 (ns 2) (n-1)/2 r(n/2) 2

f(sx' n) = (n+1) 1/2 771/2 ((n+l)sli/2 ran-l)/2)

The choice of m ± 4 • sx for the limits of the integration was chosen

on the basis that this included over

points for next year. This fact was

that computed the cumulative density

This one dimensional routine

ble integration routine to arrive at

sible point for next year's data. 'Weighting' these values by the like-

lihood of the occurrence of the point used to calculate them and in

integrating, as in Eq. 3.4, yields an XXOL for the possible new data

points.

Error Function for Probability of Bridge Failure

In order to compute the probability of bridge failure, it is

necessary to know the cumulative normal distribution. An integration

of the form:

99 percent of the possible data

ascertained from a computer run

function of f(s2 , n).

is used in conjunction with the dou-

a Bayes risk and XOL for each pos-

23

_ (n-u)r 1 s-e du

(27) 1/2 s (3,5)

could have been performed, but it was much more efficient from a time

viewpoint, to evaluate that integral using the Error Function Package

(CDC 6000 Series Computer, 1966).

To facilitate the use of this subroutine, a transformation or

standardization of variables was needed. This is accomplished as

follows:

log Q - u (3.6)s • 21/2

where division by 21/2 is the appropriate constant to yield a standard

normal transformation for the error function. The log of Q is employed

because the log of the flow is needed, since the log normal distribution

is the distribution of the annual peak flows.

Z and an error limit are the parameters passed to the subrou--5

tine. The error limit of 10 is needed because the error function is

evaluated using a Taylor's Series expansion and by specifying this

limit, the subroutine can be terminated when enough accuracy is devel-

oped in the routine.

A returning parameter E from the subroutine contains the value

of the error function, which is changed to the cumulative normal pro-

bability by

(m -u) 2

P (27)1/2 s1 2s 2

1 du = .5 + .5 • E. (3.7)

24

This is the probability of bridge failure for one year, which is con-

verted into the probability of bridge failure at least once in twenty

five years by Eq. 2.17, or in terms of the problem,

P25 = 1 - (1-p)25

= .5 + .5•

(3.8)

CHAPTER 4

SENSITIVITY AND RESULTS

The task of a Systems Engineer is not only to produce results,

but to know how these results were achieved. Knowledge of the inter-

faces between the different state variables and the effects of the er-

ror limits on program efficiency are of major importance, especially

when faced with budgetary constraints.

Choices of Error Limits for Program Efficiency

Program efficiency is defined to be adequate accuracy for the

results without too large an expenditure for computer time. To achieve

this desired efficiency, a decision had to be made concerning the various

limits that follows:

1) Limits for the mean in the double integration,

2) Limits for the variance in the double integration,

3) Relative error limit for iterations of the integrations,

4) Closeness of interval evaluation for Bayes risk,

5) Closeness of interval evaluation for the XOL, and

6) Integration limits for the possible new data.

The limits for the mean and variance of the double integration

were set to evaluate 99 percent of the area under the curve for both the

normal and chi-square distributions.

GD

2

The relative error limit for the iterations of an integration

routine applies to the successive computations of Simpson's rule. nen.

two iterations are within a specified error limit of each other, the

routine is terminated. A few combinations of these relative error limit

and limits for the mean were calculated and these results are listed in

Table 1. The final decision concerning these limits was m + 3s for the

limits of the mean, and 10 -1 for the relative iteration limit. This

produced sufficiently accurate results, while computer time was kept at

a minimum.

TABLE I

Comparison of Integration Limits

Test 1 Test 2 Test 3

Variance limits constant constant constant

Mean limits m + 2.5s m + 3.0s m + 3.0s

Iterative inte-gration limits 10 - 2 10 -2 10

-1

Evaluation ofintegration 0.9863 0.9959 1.0003

Central computerprocessing time(in seconds) 3.650 3.735 0.375

The limits for the variance are held constant in ea h case be-

cause they are the most difficult to ascertain and the 99 percent figure

was deemed 'desirable.' From Tests 1 and 2, a change in the limits for

27

the mean front m ± 2.5s to in 3.0s units effectively increased the area

under the joint probability distribution at a minimum increase in com-

puter time. From Tests 2 and 3, when the iterative integration limit

was the varying parameter, the area under the Nx - distribution is Ln-

creased to slightly over 1.0 and the time was one-tenth that of Test 2.

This represents a 90 percent decrease in computer time for each double

integration. Since the double integration is performed hundreds of

times, this represents a large amount of computer time. Hence, the

large rounding error in the evaluation of Test 3 is accepted in order

to limit the expenditure on computer time.

In the golden section searches, some type of telminating mecha-

nism had to be built into the subroutines. This was accomplished by

taking the endpoints of the 'desired bracket' and comparing their evalu-

ations for a certain degree of closeness. From trial runs it was as-

certained that the Bayes risk calculation would be in the thousands of

dollars, so a degree of closeness of $10 was chosen as sufficient accu-

racy. Each Bayes risk calculation employs one double integration, so

this is not too small a figure as far as computer time is concerned.

From the accuracy viewpoint it should be noted that a very small value,

say $1, may be fine, but since the function is very flat near the

minimum, which is known from previous trials and computer runs, this

amount of bracketing was not justifiable considering the possible repe-

tition of Bayes risk calculations that may have to be made. Also, some

additional accuracy is achieved by choosing the mini= pie depth to be

the midpoint or the interior point of the interval in question, although

the interior point itself yields sufficient accuracy for this problem.

8

The computation of the XOL employ's another golden section

search. The terminating mechanism in this case is when one endpoint's

evaluation of the 'desired bracket' is within 1 percent of the other

endpoint's evaluation. This method of measuring closeness was chosen

to correspond to the amount of error tolerated in computing the Bayes

risk, but it was made variable because the value of the XOL was harder

to pinpoint than the Bayes risk.

The limits for the possible new data points were chosen as

m + 4.0s, which accommodates almost the entire area under the 'predic-

tive' density function. This yields a range of flows between 354 and

86603 cf s. for next year's possible flows, which are far above and be-

low any momentary maximum flows observed on the Rillito Creek to date.

Actually, a smaller range would suffice, but there is no additional

time for this increased coverage, except that it may be more difficult

to compute the XOL for the extreme points, since only seventeen points

are used in computing the Simpson's rule evaluation.

A computer run, made after the original results were completed,

was initiated to ascertain the additional accuracy attained by narrow-

ing the amount of error for several of the error limits. Due to budge-

tary constraints, several limits were varied at the same time, so the

exact contribution of these changes cannot be comntified. Results and

comparisons between this run and the formal analysis are depicted in

Table 2. The amount of increased accuracy does not compensate for the

tremendous increase of 900 p3rcent in computer processing time. These

changes may be self-balancing in nature; nevertheless, the choices made

29

TABLE 2

Computer Sensitivity Comparison

FormalProblem

ComputerSensitivity Changes Caused

Limits for the mean m + 3.0s m + 3.3s Increased accuracy,less rounding error,small time increment

Limits for thevariance

constant constant None

Relative iterationintegration limit

1 0 -110 -2 Additional time for

the increased accuracy

Closeness for Bayesrisk

$10 $5 Additional time causedby a few more doubleintegrations

Closeness for XOL 10-2

10 -2 None

Limits for new data m 4.0s m 4.5s Increased accuracy,small time increment

Minimum pile depth 16.08 feet 15.63 feet Increased accuracy

Bayes risk $7450 $7451 Increased accuracy

XOL of the ori-ginal data

$2937 $2943 Increased accuracy

XOL of the newdata

$2779 $2774 Increased accuracy

Worth of addi-tional information

$158 $169 Increased accuracy

Computer processingtime (in seconds)

153 1076 903% increase in time

for the formal analysis yield sufficiently accurate results, while main-

taining a. high standard of programing efficiency.

Results

Based on the ten years of streamflow data for the Rillito Creek,

starting in 1950, the depth of piling to minimize the Bayes risk is

16.08 feet. This is equivalent to protecting against a flow of about

370,000 cfs. flow or a 'many' million year flood. The Bayes risk was

$7450. The XOL of the original data was $2937. An additional year's

data showed an XXOL of $2779. Thus, the expected value of one more

year's data is only $158.

The cost of obtaining one more data point is the actual cost

of getting the data point, plus the loss of benefits from the bridge

for that year. The total cost for building the bridge to the design

specification at present is $156,440.

Sensitivity of Objective Function

A faunal sensitivity analysis is meant to show the effect of

changes of pertinent parameters that are embodied in the objective

function. The following variations of these parameters will be dis-

cussed:

1) Changes in the sample size, or selection of different

years' data,

2) Selection of a different underlying distribution for the

data other than the log normal,

3) Changes in the costs for the bridge and piers, and

4) Changes in the depth-discharge relationship.

31

Sample Size; or Selection of Different Years Data

The choice of 10 years of data wus made. arbitrarily. The

latest ten years of Rillito Creel: annual peak flow data was employed,

but a change in the results was found to occur when a different ten-

year period was used. The period between 1940-49 poPs- sed a higher

variance than the 1950-59 data, so it is expected to exhibit a higher

Bayes risk and EVSI than the 1950-59 data, because there is more uncer-

tainty caused by the larger variance.

Analyses were also made on different sample si:es, particular-

ly five and 44 years of data. In the case of five years of data, the

worth of additional data is more valuable than for ten years, which is

to be expected since it is a small sample size subject to larger fluc-

tuation when additional data points are added.

The results for 44 years of data are interesting from the point

of showing the roundoff error involved at the accuracy level specified

and Herfindahl's (1969) diminishing returns of additional infolmation.

The EVSI can never be negative by the definition of this problem, in

that the expectation of next year's possible data points is based on

the present data; hence it is impossible to get an EVSI less than zero

because the choice of points and their associated likelihood cause a

value of EVSI greater than or equal to zero. The computer program

yields a result of - $3.00 for the EVSI for 44 years of data, which is

necessarily caused by roundoff error in the program.

A summary of the above results is presented in Table 3.

TABLE 3

Comparison of Different S3mple Si :esor Different Selection of Years

Formal1950-59

10 Years1940-49

5 Years1949-53

44 Years1917-60

Mean 3.750 3.597 3.758 3.696

Variance 0.070 0.138 0.032 0.095

Bayes risk $7450 $10089 $12994 $6064

Minimum 16.1 feet 20.3 feet 13.0 feet 14.0 feet

XOL $2937 $4542 $6730 $1613

XXOL $2779 $4241 $5513 $1616

EVSI $158 $301 S1217 -$3

Selection of Different Underlying Distribution

The selection of the underlying distribution can be the cause

of a large change in the Bayes risk and other calculations. The objec-

tive function V(h, u, a 2) is sensitive to the underlying distribution

in the following manner: P25 is the probability of bridge failure at

least once in twenty-five years, which is a function of the underlying

distribution. The lower the pile depth necessary to ascertain a fair

.amount of certainty that the bridge will not be destroyed by bridge

failure, the lower the cost of constructing the bridge, due to less

money spent on sinking the piles.

From Fig. 5 it can be seen that at point dsi on the horizon-

talaxis,theprobabilityofbridgefailurecompute issi

cls.

Fig. 5. Cumulative density functions for differentunderlying distributions for the streamflowdata.

33

smallest for the exponential, next smallest for the log normal, and

greatest for the uniform distribution. Thus. , if these values are in-

serted in the formula for P25 and computed, the value of the objective

function is largest for the uniform, followed by the log normal and

then the exponential distribution.

Therefore, the choice of the distribution for the annual peak

flow data has a definite effect on the cost of constructing the bridge.

Changes in the Bridge and Pier Costs

By careful observation of the relative costs of the objective

function, it becomes obvious that the best strategy to follow in con-

structing the bridge is to sink the piles deep enough to almost guaran-

tee that the bridge will not fail. This incurs a cost of $400/foot for

sinking the piles, plus the bridge cost; but there will be very little

risk of the bridge failing and hence little chance of having to replace

the bridge. This makes the formal analysis used in the thesis somewhat

superfluous in that it really is not needed except to save several hun-

dreds of dollars, which is negligible when compared to the actual con-

struction cost of the bridge.

If the structure and pile costs were closer to equality than in

this thesis problem, the formal analysis would be of great value in in-

hibiting the cost incurred by the conservative who overdesigns or the

'gambler' who underdesigns in order to 'save' money. This could be the

case for an oil rig in the middle of the ocean, where the cost of sink-

ing the piles would be much more in line with the structure they are

supporting. The additional cost of sinking the piles unnecessarily

deep would not be incurred if the approach used in this thesis were

employed.

Change of the Depth-Discharge Relationship

Laursen (1969) has stated that this relationship can be de-

picted using a power function to the 3/5 power. This results in the

empirical relationship shown in Eq. 2.13. A change to a power function

to the 4/5 power causes the exponent in Eq. 2.13 to increase to 4/10,

which means that a larger flow is associated with the same pile depth.

This is then reflected in the cost of constructing the bridge because

the larger the flow protected against by the pile depth, the less pile

depth needed; hence a lower construction cost. The reverse argument

is true for a smaller power function.

alum s

CONCLUSIONS

The low value for the worth of additional information is in

part due to the relatively flat minimum of the obi active function for

the Bayes risk. For the pile depth in the range of 14.88 to 16.82

feet, the Bayes risk varies by no more than $100.

Clearly, the bridge should be built now, on the basis of the

present knowledge because very little is gained by waiting for more

data.

The computer program takes only 153 seconds to arrive at an

answer, making this fofm of analysis very inexpensive at the accuracy

levels shown in Table 2.

Design modifications, other than those discussed here, are easi-

ly handled by this method of analysis. Changing of the objective func-

tion and other variables still yields the same basic design principle

-- choose the design with the lowest Bayes risk, if building on the

basis of present knowledge.

The value of additional infolmation can be expected to change,

based on the variability of the design parameters, especially the number

of years of data that are used in the analysis.

36

APPENDIX A

LISTING OF VARIABLES

lower limit for integral

upper limit for integral

BC bridge cost in dollars

2X a percentage points of the chi-square over degrees of freedom

adistribution

d flow depth in feet

df degrees of freedom

error function parameter

constant

EVSI expected value of sample information

g(m) lower limit for integral

r(x) gamma function of x

h pile depth in feet, depth of scour in feet

h* optimum pile depth for unknown u and a2

ht optimum pile depth for known u and CI

2

h* optimum pile depth for known u and G 2 with additional data point

h(m) upper limit for integral

number of years

37

38

constant

constant

In sample mean of peak flow data

MXsample mean of augmented peak flow data

sample size

N2 (z) convolution of normal and chi-square density functions

OL opportunity loss

probability of bridge failure in one year

P(y) binomial distribution of y failures

P2 5 probability of bridge failure at least once in 25 years

flow rate in cubic feet per second

R(z) Bayes risk

sample standard deviation of peak flow data

s 2

sample variance of peak flow data (biased)

s 2

sample variance of augmented peak flow data

2 population variance of peak flow data

population mean of peak flow data

V(z) objective function or utility function of vector of parameters

z, equals F(z)

VSI value of sample information

XOL expected opportunity loss

XXOL expected expected opportunity loss

y number of bridge failures

APPENDIX B

GENERATION OF NORMAL CHI-SQUARE WITHCONCOMITANT UNCERTAINTY DISIRIBUTION

This distribution falls in the family of distributions labeled

by many statisticians as 'predictive' distributions. The development

of the following approach was accomplished by Davis (1971).

Reference to a normal gamma-2 distribution was made earlier in

this thesis. This distribution is necessary for the development of the

normal chi-square with concomitant uncertainty distribution.

Before proceeding further, a list of additional notation is

furnished for use in this appendix:

L[x] likelihood of x,

proportionality constant - 1/G2 ,

sample size,

in sample mean,

y sample variance,

mX

sample mean with additional sample data,

s 2 sample variance with additional sample data

population mean,

2a population variance,

df degrees of freedom,

NK[x] non-kernel of the distribution of x,

domain of the function

39

fN (x) normal density function of x,

(x) gamma-2 density function of x,fG2

vector of uncertain parameters,

E (W) probability density function of uncertain parameters.

To begin, a statement of Bayes theorem is,

_ (W) f Ex 1W) (BI)f (x IV) (W) d117

where x is the new sample information.

The predictive density function is being sought and ifWwere

known, the predictive density would be f(x111). However, only the pro-

bability density function of W is known. Thus, for any specific x, the

predictive density function evaluated at that x is a 'weighted average'

of the f(xIW)'s, with each f(x1W) being weighted according to the like-_

lihood of W. This is accomplished by the following integration,

Sf(xIW) (-6T) dW

(B2)

which is exactly the denominator of Bayes theorem. Thus, rewriting

Bayes theorem yields:

Jf (x IIV) (w) dw - (W) f (x 1W) (141x)(B3)

11

Note that the numerator and denoiT-Iator are known because of the

conjugate relationship, as shown by Raiffa and Schlaifer (1961). Hence

the kernels, or portion of the likelihood function containing the vari-

able parameters of the distribution, of the numerator and denominator

are exactly the same, which reduces the problem to:

f (x1W) NK OV.) °NK f (x IV) NK E,- (Wix)

The normal gamma-2, as described by Raiffa and Schlaifer

(1961), can be written as:

fNG2 (u, km, y, n, df) = fN (ulm, kn) • fG2 (kly ' df) . (B5)

The likelihood of the normal portion of the normal gamma-2 is:•

1 - 1/2 1m(n-u) 2(1m) l/ 2L[fN (uirrl, kl1)] - (27) 1/2 e

(B6)

so the non-kernel portion, or what is left after the kernel is extracted

from the likelihood function, is:

NiqfN( , kn)] = (n/27) 1/2 . (B7)

The likelihood of the gamma-2 portion of the normal gamma-2 is:

Jvkdf vkdf/2) 2 (vdf/2)L[f ( 1 , df)] 7 e (df/2-11)! , (B8)

so the non-kernel portion is:

(B4)

NK[fNG2 (n )] _ (vd1/2)df/2(df/2 _ 1), (a/27) 1 2

Since df = n - 1, Eq. B10 can be written as,

1/2 ((n-l)v/2) (n-1) / 2Niqf

[G2( 1)] = (n/27) ((n-1)/2-1)!

After taking a new observation, Eq. Bll becomes,

(ns 2J2) n/2NK[fNG2(11+1)] = ((n+1)/27)1/2 x (n/2 - 1)1

vd ,df/2('i'' •NK[fG2 (k1v, df)] =(6t/2 1)!

Combining Eq. B7 and B9 yields the non-kernel part of normal

gamma-2 distribution, which is,

Now Eq. B4 becomes

0 1 2 an-l)v/2) 0-1-1)/2(n/27) ((n-1)/2-1)! f(xIW) E(11) dW - 2 n/2(..)

(0a+1)/27)

1/2 (nsx/2)(n/2-1)!R

where (1/27) 1/2 is the non-kernel of the normal.

Eq. B13 'reduces' to,

(B13)

42

(B9)

43.

Li1/2 ((n-1)v) (n-1)/2r((n-2)/2) (B14)f(x1W) E,(10 dW = (1/T ) 1/2 (n/(n+1)) ' • ,

((n+1)Ç 1141 ' r ((n-3)/2)

Finally, substituting ns2 = ( a-1)v into Eq. B14 yields,

(x 1W) (W) dW = (1/ Tr) 1 2 2,(n-1)/2 r((n-2)/2) (B15),,,

2.01/2 ((n-3)/2)(n+i) ) 1/2 )

an-1-1). sx)

which is readily solvable in terms of the problem.

APPENDIX C

LISTING OF COI\IPUTER PROGRAM .ANID RESULTS

This appendix consists of the actual listing of the CDC 6400

computer program, followed by the computer results. It was written in

FORTRAN II and TV programing language. An explanation for the func-

tion of each routine is supplied via caRment cards at the beginning of

each subroutine. Further comment cards attempt to delineate the pro-

cess that is being perfolmed at each section in the subroutines.

The output is for the formal analysis of the problem using the

ten years of streamflow data for the Rillito Creek.

44

45

. .-,.,'.' .'_ ,,'.....) \: ,_):-:(1.i,--'1,',1 0uHul,Hk.)..,,C, r)

4-r,

* i' ',L ":ii ." ,,'—'. f' ' ',..' :....*-',TE

G

:... , 1Lc. , 1 ,,..? r 6H106L -,,SJ- H.! 1 i,-. - .. , i'»,_.!., 'P(ko ,

TH-12 Fi..2.LLL ,., L.,, ,J1-.. -, , -„L : -';1'.--_-.i,-,L ,---- [-'-- E »FHI

ILG ,i ,..):: . ,-*10'H'-, iv70, G4. ** .,,=,):... Cz6-.:DS Fi.)LLo-iii THE F ,A_Lc-iIL -a FOki,!--..1 ...-- di

C. ..J.,..---),L,:::: 1I,T G* (-,:..)L 1 - ai, -.) '--E.Ai_ ht:-/-ADIkf.., Fok uUTP1

C.,.,L).::-. coL 1 - 16 CH1 .1.--'1d C .,.)T ir* CJi,._ fl 2:_) CP? br;10(,:ii:-. COSf G* COL 21 3 f_. k 0-icou;.,.11 Fx'ACHk .0.* -,;KO 4 ** COL 1 - 5 -JDF-,TS ki_i_:_,J.,'R i_J4%T. ,- HOli,'..ls *

* C.:0i, 6 - 15 FHT vi,i-zIAHLL IJAIA Fu,---,( --;;Ai ** CUL 16 - 20 ;..:Yk STr:,,h211N0 i.JiAri-\ YcLir.r. ** CAPO 5 ... ..,.0.. .0-* DO17, ** *G ***************G0**************************G***************

CuW10p4 // -/c(1V) tALOG(1u0)/5EAH/H9CP2,Cv1s905

CO(10;9 / 5 I-IP/4HEA 9 , AkNDPTSs.1uF,CO;;00:'4 /COLAj/IIIIJIMENSION HEAD(32)9LIE(135)1)!\TA 4.1EAHoYAH/2*./ ,LIE/1311 -1*/

11 11 =H[7::AD THE I:PUT OR DATA CARDS

HEAD 21HEA1)HEAD 3,CP19CP29RREAD 49.NC)PTS9FtAT914Yis,

READ FMT,(A(I) 9J=19hDRTS)C RITE IhPUTED DATA

PU'!T 50HEAO,CPI,CP29P9NYi:9(/\(I)911,NOPIS)TAkE L065 OF FLowS Af'4D STORE IN 6AmL AkkAY - A

oL) b I=10:0PT5XL06(1) = AL0610(<(T))

7,(XLOG(I)91=1,DPIS)C LOtAPUTE HEAN AND VAH1ANCE FOR-Lou FLoieS

00 8 1=1,,UPTS8 AP1EAN /J.1EAN

XL06( 1 )/NOPTSDO 9 I=1 ,0DPTS

46

v J V" +( (0( 1-) — ',', )*(.(I) — NICc ..)A / ippl5ST,i)V = F,T(V,F-...LJ l', I PADPT ,.„ 9 ,,if

C f:.,c,APuTE5 GijiHA FU -4CTIiiA- = Nt)PTS - 1III =. :OFTAIDFq?)IF (III.) 4 :,;9.5.943 .

4 , 6i01 = GA(ILJ)6 ..: TO 6C

5.; 6An = G4J-1 I(IDF)o c CONTIuL

C LOTIvE CUnTPOL OF PkOGFn FL0 .6CALL CSEAr:CH(STnEv)CALL SEAPCh(STLIEv ,;TU)C.ALL ,v,PT(T00)DI &f = To _ fooPRIJ“ 219!,HEA:05LIHEILINE9LINE,TO,Tu 0 90IFI:. 4LINE9

A LIE,L1hEC FORMAT STATEHENTS

2 rol.T(16A5/1(DA5)

3 FRnAT(3Fl.c.)

4 F0kAT(15A115)

5 For+AT(1H1,16gb/6xIloA5////1149*FIrt co31 = * 9

AF1).2/119 *BkIOGE 0 (3 5f F *,FP.)0/11A9*DISC 01,1 11 *

A* = *,F1.,;e2///7/6s*DATA E - GINS ,i YEA ,I)////C21X,*--- DATA PO“\TS ---*//// (24X9F10.3))

7 FR1AT(ihi,13A9* LOGARITHmS • *9///(11A9E20.8))

I..; FUR,IAT(////6X,*--- nE/',N Ar40 VAHlAfACEI FOR 5 A0r)LE *

A 0 0 .F SIZE *,15//////16Ao*vEAN616x9*vARIANCE =*,F15,7)

219 FORmAT(1P1f//A91645//6x916A5/////4 ,1 3 5 A 1 /X9 135 A 1 /

A,135A1///X9*FINAL RESULTS OF STuOT*////A,6*EOL OF ORIGIAL OATA =.23.6//x,CoEOL OVER ALL AE,4 RUSSIELE POINTS =*,E11.6//////

- * iciRTri uF r_1) ., E nORE 0A1A POINF =*

EE15.7///X,135AI/X,1 3 541/X91 35 A1)

STOP .Ei4U

47

FUNCflON GAMI(1)**************i******************************************

COHPUTES THE Gi ,'4i/A FW-CT1UN FOR INTEEliS

**********************************************************

GAM1=100.1 J=19K

1 6.4M1=6AHr*j't - ETUkNEND

FUNCTION GjOY(I)**********************************************************

COMPUTES GAMMA FUNCTION FOP NON INTEGERS

**********************************************************

(I,E(4.1.0R.I.EQ.3) GO TO 3K=2*(I /2....65)°1GAMR=1.00 1 J=I9K92GAMR=GAMk*(0/2.)GAMR=GAHR*SORT0.1415927)GO TO 4

3 GAMR=SOT(3.14159a7)/2.

4 RETURNEND

48

sooPOLATIE CiSEACH( (,,Tf , LV)**********************************************************

SE,4PcH t-.)Y (;OLDEN 5ECT .iw , ITCkkTIVE PkOCFSSSEACHES FOc- OF FU0CJI0 14

kEFERECE vILOE ANo HEI(--,1TLERFOODATIONS OF OPTImIZATION

F- Ck EACH PILE DEPTH CHOSEN THE ROOT“iE CALLSIMP5u, :;HICH IS THE 00UoLE InTE6ATIOr4AL.) THIS IS ThE CfALCuLATION OF THE bAYLS RIR!.4HICH YIELUS THE HIlmUH.

**********************************************************COMMON /SEAk/H9SC,PC,XMIPA,DSCOMiloN /SINP/X8AR9vAk90O0,1016AmCOWION /CCUNT/IIIIOINENSIOh PT(4)9FV(4),LIE(135)DATA CLOSE/1,/,LIr4E/135*1H*/

DEPTH TO FLo',y RELATIONSHIPr(Z) = ( / .344 ) ** ( iS / ' I

PRINT U:,0t),LikEINITIALIZES POINT .f)NO SIZE OF JumP

=PT(1) = 9.8SS = 1,GJUMP = 1.2*S5

SETS PILE DEPTH, CONv ERTS TO FLOw, Lo6 OF FLO,AND EVALUATES VIA THE DOuBLt-- UITEGRAFION_

OS = PH')= Q ( P 1 (1) )

H = AL061,(H)CALL SIMPSON ( EV(I) )PT(2)=PT(1)+6JUP

C SETS PILE DEPTH, CONVERTS TO FLOW, LUG OF FLOW,AND EVALUATES VIA THE DOUBLE INTEGRATION

DS = PT(?)H = Q ( P 1 (2) )H = ALOGI(H)CALL SIMPSON ( EV(2) .)

CHECKS EVALUATIONS FOR OIRECTION OF NEAT JtJi-YIF (Ev(1).GT.EV(2)) 1,4

SETS PILE DEPTH, CONVERTS TO FLOw, LUD OF FLU,Aû EVALUATES vIA THE 00QmLE IN1..16RA1I0fJ

1 RT(3)=PT(2)+1.b18.,34*(PT(2)-PT(1))-0S = PT(3)H = Q ( P 1 (3) )

H = AL061,(ti)CALL SIMPSUN ( EV(3) )

C! -IECS EVALUATIO:9S 'A)H kE.O ,;,.oEkio OF FOJNisIF (FV(3).GTePv( -, -. )) 1193

3 PT(I)=PT(2)P 1(2)=I- T(3)LV(I)zEv(2)EV(2)=Ev(3)GO LO I

C REORDERS POINTS

4 TA = PT(I)TB = EV(1)PT(1) = PT(2)EV(1) = EV(2)PT(2) = TAEi(2) = T8

8 PT(3)=PT(2)-1.618,:34*(PT(1)-PT(2))CSETS PILE DEPTHi CONVERTS TO FLOeJ, Loo OF pLo ,:J 5,

ANO EVALUATES VIA THE 00L-LE IiNTEOPATIONOS = PT(3)H = Q ( P1(3) ).H = ALOG10(H)CALL SIMPSON ( Ev(3) )

C -CHECKS EVALUATIONS FOR REORoEING OF POINTSIF (EV(3).6T.EV(2)) 22997

REoPDERS POINTS7 PT(1)=PT(2)

EV(1)=Ev(2)PT(2)=PT(3)EV(2)=EV(3)GO TO 8

C CHEcKS TO SEE IF EXTERIOR POINTS ARE ITHIN

C - SPECIFIED LIMIIS11 = AbS(EV(3) E4(l) )

IF (w.LE.CLOSE ) GO TO 2 026 PT(4)=1.616034*(PT(2)-PT(1))+PT(1)

C SE1S PILE LEPTH9 CONVERTS TO FLo'd, LUG OF LOw,

C AND EVALUATES VI, THE D0u6LE INTEGRATIONDS = PT(4) .H = 0 ( PT(4) )H = ALOG1((H)CALL SImPSON ( EV(4) )

C DELETES FATPANEOUS POINT BY RE0kDERIIY,5IF (EV(4).L.T0EV(2)) 1019102

c REORDERS POINTSlui PT(1)=PT(2)

EV(1)=Ev(2)P 1(2)=PT(4)EV(2)=EV(4)GO TO 11

C REORDES POINTS1a• PT(3)=PT(4)

49

50

EV(3)=EV(4)GU TO 22,- '

C RFORDEPS POINTS22R T=P1(3)

TT LV(3)RT(3)=PT(I)EV(3)=EV(1)ri (1)TtV(1) = Ti'

C CHECKS TO SEE IF EXTEKION POINTS ARE ITdiNSPECIFIED LIMITS

2Lv i = AGS(F1(3) Ev(1) )IP (‘.•:.LE.,CLOSE ) GO FO 26;_)RT(4)=PT(3)-1.618f)34*(PT(3)-PT(2))

C SE IS PILE DEPTH, CONVERTS TO FLU, Lou OFAND EVALUATES TdE DOUBLE INFL*RAW.4'4_

OS = PT(4)= Q ( PT(4) )=. ALOG1(H)

CALL SI('!PSON ( FV(4) )C DELETES EATRANEOOS POINT bY REORDERING

IF(EV(4).GT,EV(2)) 'C'01,202 •2'21 PT(1)=PT(4)

Ev(1)=EV(4)GO TO 11

C KEORDERs POINTS22 HT(3)=PT(2)

L-_,/(3)=EV(2)FT(2)=PT(4)EV(2)=EV(4)GO TO 22(J

C CONPUTE'S ANIN ENUAL TU THE NrDPO1T OF INTERVALA;II0=P1(1)+ .5*(PT(3)-PT(1))

C SEAS PILE DEPTH, CONVERTS 10 FLO, LOG OF FLOvq,ADO EVALUATES VIA THE DOUBLE INTEGPAIION_

DS = AMINH = 0(XHIN)H = AL061(H)CALL SIHPSON(V1)

C COMPUTES BAYES PASK AND MINIMUM PILE DEPTHVi = AmInl(V19Ev(2))1F (V1.EGI.EV(2)) XMIN = PT (2)

C PRI'rr RESULTS FOR 3AYES RISKPRINT 22,LINE,AmIN,V19LI1'E

22 FORHAT(////X,I35A1// X* RESULTS FOR bi.AYE*A* CALCULATIONS */////6X,*mININLP1BET.8//6A,*BAYES RISK =*,E20,6///A,13541)

lo0(-2 FOHMAT(1h1/// X,*JOuBLE INTEGRATION TO FILD ThE *A*NINIOUM PILE OEPTH*/3A, 0 0vER ihE 0OUDLD INTEur“„ *

B*TION *///X135A1///)RETURr4ENO.

5 1

StiL5P.OUTIl'iF SIP3Oi“(01)* *** **-; 1- ***********4************ *********************

COTROL PROGkAm FOP pbuLESETS THE LIMITS FOR TriE (7)0TC.k 1.1E1.,J, URVAkIL:CL n4 THIS CAbE t AU IS PROG1-00 To okL,ANThE Coi UAkt DISTIHUTION OP toJu 1H:L1 SEGHENTsr-oR FASTER INIEGRATION 1.4 0CEEDINGS.

******* * * ** *** * *** * *** ****** **** **** ** *** ******** *********LCflr'Oi. ,; /S1( 1 P/u9S941UF/Cv'tCuMrUH /EAR/h96CsPC.;X1IL ,,DS

/0000T/ 1111L)1HEJ,SIWJ CH11(44)1,CH19(44)

LOAUS ViALOES FOI . CHISJOgRE SO TmATCAN 0 E ESTABLISHED

DATA CH119CHI9/.6s4 , 6G14.fJOis.C2279.429.3b?A.k_.6541.1C719.128190147916679.14b;.2..)134.21(2v6.23 -224.24644,25989.2725281-6,2019.3u(4.31(44C..32739 .3 -3099.346,.35q79.76319. 3 7119.378o9,, 3d.539,U3934,. 44904C7, 9 4134T6,4197,04257,. 43159043711449E..44-73,

ti) ,"'F--:1 5, 4+Q 103; 3 G 71-03.476,i3ei:b56 9 3.u97 6:.,2.95r3o962.8422,2.7424 t 2.65o,2.52,2‘.513192. 4 53292.3 94 ,

: -y..3,5792.306392.265692.226492.1949.1 6 2192.1325912•1046,2.Y789i2.6547q2.0319v2.01049 1 .99:191.9/ 09

01.95 2 7,1,935591.9194 1.9V3 ,4.91b8 3 55, 1.074291.bbCosKI.8476,1,83591.E.2391.811.5,1.60L4,1.7696/TuT =

SETS UNITS FOR A AND o FOF FIRST HALF OF INTE6RAfIONA = (t-i*S) (IuF*C019(IDF))

= (r, * S) / ( IDF 2)CALL SRULl(AOIAREA)

C ADDS INCREHETAL AREAIOT = TOT 4. ii-(EA

SETS LIIiITS FOP A AND b FUN SECOND HALF OF INTEuKATION_A = (t S) / (IDF — 2)6 = (N*S) / (IOF*CH11(IDF))CALL SuL1(A,69APEA)

ADDS P,ICREt;ENTAL VALUE, OUTPUTS RESULTSTOT = TOT 4, AREAIF (IIII.E0.1) RETUR N

1(;C.059 TOTlc° FORMAT(lh,,,X9*PILE DEPTH =*,F1U.24n44

A*PISK CALCuLATION =*9.E16.7)PEToRe,ENV

52

FUNCTION PHI(5kR)***************************-r** *************************

FUCTIO ROUTINE E.:EOED TO COhBINE THE Tv40ONE-OlvF.NSIONAL SLIPSONS RULES. - RESETS THEUNI 15 ON THE INTEGRAL EACH TINE I] IS CALLLO.

-**********************************************************

A=G(AG)= hA(APG)

CALL Sk.UL2(A.6,ARGIRd1)1F(SEVSE SITCH 1)111,222

111 RRINT 77.ARG9PHI77 FLWi1AT(51-1 ARG= F20.6.10X4HPHI= F20.6)

222 RETOPNEND

FONCTION HX(X)4*********************************************************

FUNCTIOA ROUTINE TO SET THE UPPER LImiTS ON THEINNER INTEGRAL F(,) EACH POINT OF OUTER INTEGRAL.

**********************************************************

COt'HON /SIhP/4M,V,HU9109GH 1 + *1.0 * SOkT(A)HETUNNENO

FUNCTION G(X)**********************************************************

FuNCTION ROUTINE TO SET LO W ER LIMITS ON THE INNERINTEGRAL FOR EACH PINT- OF OUTER INIEGI-e.AL.

4

**********************************************************

COMMON /SIMP/XM.VIOD,ID,r;AllG = Xm - 3.0 * SORT(X )i- ETURNEA)

SUbkOOTILE SPUL-2(AAkOVAW:',A)****************************************************

A OAE DIHENSIONAL SIjPSON NULL THAT C0HUTESv . i .A-UE 0F iHE fATF6AL Q LP OHPU1FS fhEPoINT ,10PE TH AA ONCE. INSTEAD CHAN6ES THE* C0EFFIC1ENT OF THAT TER".

•

**********************************************************C SETS AND CHECivS LPIITS AAD INTEPVAL SIZE

DELH=(6—A)/2.IF(DELH)"2:9192AREA=.(.40 TO 70

C CALCULATES VALUES AT EXTREHE POINTS2 AIN=k(APG,A)*P(ARG90)

C iLXT POINT AND EVALUATESOltg#DELHYl=i--((59C)PRAk=DELH/3.*(APIN+40*Y1)

C CALCULATES REPAI;4I ,AG HOINTS AN0 cHAN6Ls COFFFICICOF THE PAST POIFJTS DURING THE CUHPUTATIOH

DU zH) 1=2/15N=2**(I-1)FK=KAREA=APIN.2.*Y1K2M1=2*K-1FL=-1.

DO 6:J H=K,K2H1FL=FL.c?..C=A+FL*DFLH/FK

60 Y2=Y2+R(ARG,C)AREA=(AHEA+4.*Y2)*DLLH/(3.*FK)YI=Y1-PY2

C RELATIVE EPROREPS=AbSF(AREA*1.E-61)

C CHECK TU SEE IF 1-v10 ITERATOHS ARE 4ITHIN 10 PERC&JIF(ABSF(AREA—PRAR)—EHS)70,71,)940

4u PPAR=AREk70 FETUP.1 ,1

END

53

54

SUBF:00TIA SkULI(A9h9AkEA)**i,***********.************************.****************

ir

A oilE D1ENSIONAE SINPSON iAJLE THAT (:Jr,,PurEVALUE OF THE INIE6RAL. 6EVER COMPUTES tHFPOINT ONE THAN ONCE. INSTEAD CHAi:uES THECOEFFICIENT OF THAT TERN.

********************************************************SETS ANu CHECKS LIMITS AND INTEPvAL SIZE

DELh=([3;-.4)/2.IF(OELH)29192

1.6C.) TO 70

CALCULATES VALUES AT EXTRE` , 1E POINTSAkIN=PH1(A)+PHI(k)

C NEXT POINT AND EVALUATESC=A+DLLPYA.PHI(C)PkA=0ELH/3.*(APIL+4.*'(1)

CALCULATES REMAINING POINTS AND CHAN'o'c_S COEFFICTEI,JSOF THE PAST POT!TTS DuHING THE CONPUTAFIONS

00 4(. I=2,15K=2**(I—I)FK=NAREA=ARIN4-2.*Y1K2M1=2*K-1FL=-1.Y2=J.o0 e 11=K9K2t;i1FL=FL-1-2.C=A+FL*UELH/FKYR=Y2+PHI(C)AREA=(AKEA+4.*Y2)*DELH/(3.*FK)Y1=Y14-Y2

C RELATIVE ERROR LIMITEPS=AHSF(AREA*1.E-01)

C CHECK TO SEE IF TwO ITERATOhS ARE WITHIN 10 PERCEr1TiF(ALISF(AREA—f:RAR)—PS)7L,97U,40

4C PRAR=kREA7., RETURN

END

55

FUNCTION R(X9O)**** 14. ****************************************************

i-‘

0

FUNCTIOk EVALuATFO IN 00 1_16LF INTEGA(iu is RpFOR HAYES RISK iT kE1UP'S AFTE CuHRuTAITON 4NO00NrIHHES PROcF_SSIIN6 ()SING THE uuLLIEN SE(;110i , .

****************************************************ti-*****e0M01, /PRP,R/RCU' Oh /SIOP/A64S.N.Iv96AllLoHdUN /COUNT/III1 -

/SEAR/H9F,C9PC.XhIDSCGMON /NINImo/TOIOENSION P(4)9FV(4)DATA EPS/.00E)1/

C NILE DEPTH TO-FL(Pj RELATIONSHIP= ( Z / 4344 ) ** (13e / 3.)

L1KZLIHOOD OF NORMAL OhI-Sc,luARE 0IS1K160TioNRk= ((N * s / yo **((iV-)/ 2.) * H * s / (A *

A X))) / ( GAO * (2. **(Ivi 2)) * ST(5 1,2331b34H * x) * FxP(( * S / (2. * X)) 4- ( ( J - AS) *C - X13)) / (2. * A)))ELL. =IF (IIII.EW.1) GO TO 30

C NO1-61AEIZING FACTOR FOR LOG OF FLU',,4Z = ((H-U) / SORT(X)) / S(_,)RT(2.)IF (Z) 10109162041b20

1 -J10 Y -= AdS(7)C CALLS ERROR FUNCTION TO EVi-LUATE NORNAL PRoBAbILIFY

CALL ERF(Y9EFS.E.Fid.C)cOHPUTE PROBABILITY OF BRIDGE FAILURE Jr ONE YEARRP = .5 + .5 * EIF (PP .GT. 1.0) PP = 1.0GO TO 1°36

lj.i2o CALL ERF(Z9EPS9E4HH9C)RP = .5 - .5 * EIF (kR 0.0) Rp = O.

PROBABILITY OF 6RIDGE FAILURE Ar LEAST ONCE IN 25 yks103 0 P25 -= 1. - (1. - PP) ** 25.

C EVALUATE THE OBJECTIVE FUNCTION WITH THE EIKELlhOu0

• I H = RN * (P25 * BC + PC * US)RETURN •

C SEARC"( BY GOLDEN SECTIONC INITIALIZATIONS OF JUNE'3 n., ST = SORT(X)

GJP = 1.2 * ST

56

C CU:.IPUTES POINT. THASFORN FLu5i. LOG Or: FLU 5AND EVALUA)ES NSIWG A INrE6Af11,,N

P (1) = TC CO.iPUTES NP5 POINT. TP/ANSFON 1LJ, LO ( , OF FLW -25

AND EVALOAlEe, 1 ;1N(-) A INr ,- (3RATION PrWcEOuKEDS = P(I)IF (OS *Li'''. 60 10 2H = OW(1))EV(1) = HT .VAL(X9U)P (2) = T GJPOS = P(2)IF (DS .LE. Gu TO 2ti = Q(F(2))EV(2) ' -VAL(xu)

CHECKS FOR DIRECTION uf- NEXT 000.)

IF (EV (1) .6T.EV (2) ) 1 6 9 4C CON PUTES ;E P014 T000SF0k0 FLU LU 6 OF FLD.

AND EVALUATES DSIi40 A 1NTE6RATION PKOcEDURE0 (3) = P(2) 4. 1.638334 * (P(2) - 0 (1))DS = P(3)IF (DS .LE. O.) GO TO 2H = O(P(3))EV(3) = HTVAL(A9U)

C CHECKS FDR DESIRED bRACKET OR r4,70 P34 (0 BE AWOL')IF (EV(3).GT.Eçt(2)) 1253

kFORDERS POINTS SO OLv , POIT CAN BE WJDEO3 0 (1) = P(2)

P (2) = 0 (3)EV(1) = FV(2)EV(2) = EV(3)GO TO 10

C REORDERS POINTS $o f4 CAN BE AUDED -

4 TT = P(1)TTT = Ev(1)P(1) = F(2)EV(1) = Ev(2)P R) = TTEV(2) = ITT'

C CoHPUTES NEw POINT , TRANSFORM FL04, LUG OF FLO'ii.040 EVALUATES USING A INTEGRATION PROCEDURE

8 P(3) = P(2) - 1.613034 * (P(1) - P(2))DS = P(3)IF (DS • LE. C.) 30 TO 2m = u(P(3))EV(3) = mTvAL(Xtu)

C CHECKS FOR DESIkED f7,'0ACKET OR NE-i POINT TU BE ALOEIF (EV(3).GT.EV(2)) 22997

REORDERS POINTS SO NEt, POINT CAN SE ADDED7 P(I) = P(2)

P (2) = P(3)Ev(1) = EV(2)

57

CC

12

11

EV(2) = EV(3)GO TO ROOTIi4oE

CHECKS TO SEE IF EXTEPluk Oi -7.SPECIFIED LIIIS OF 1 PU,T.ET

= A 6 IN1(EV(1) , Ev(3))

'AJTHIN

YY = A6S(EV(1) - EV(3))AX = .01 *IF (YY .LE. XA) GO TO 20:)

C HUT CLOSE - FAOOGH SO hoD ki_A POINTC CO POTES POINT, 1k4HSFot-1 F.L09 Lj(i OF FE09C Afsiu EVALUATES uSING A PATE6r‹,:0- 10H PkOcooL26 = 1,618034 * (P(2) - P(1)) P(1)

JS = P(4)IF (06 .LE. 0.) GO 10 2H = O(P(4))EV(4) = HTVAL(A90)

C OELETES EATRANEOOS POINTIF (Ev(4).LT.EV(2)) 1019102

C .0LoRDERS101 P(1) = P(2)

p(2) = e(4)Ev(1) = ETV(2)EV(2) = F0 (4)GO TO 11

C- C

DELETES EXTRANEOUS POINTriF_ODERS POINTS

1 02 P(3) = P(4)EV(3) = FV(4)GO TO 220

C HEORDERS POINTS229 TT = R(3)

TTT = EV(3)P(3) = P(1)EV(3) = EV(1)P(1) = TTEV(1) = - TTT

C CHECKS TO SEEIF EXTERIOR POINTS ARE vJITHINSPECIFIED LImITS OF 1 PERCENT

220 N = AmIN1(EV(1),EV(3))= ABS(EV(1) - EV(3))

XX = .01 * W1F (YY .LE. XX) GO 10 200

NOT CLOSE ENOUGH, SO ADD NE! POINTLOAPUTES •FW POINT, TkAHSFOPH FLuA, LOG OF FLU,

CAi) EVALUATES USING A INTEGRAflu.4 PRoCEOURLP(4) = P(3) - 1.6161;34 * (P(3) - P(2))

= P(4)IF (DS .LE. U.) GO TO 2o = u(P(4))EV(4) = HTVAL(X,U)

58

IF (Ev(4),GT.E.J(2)) 201,202C 0ELETES EATRANEous 1-)011qC t:EORDEqs P01 MIS201 P(1) = P(4)

EV(1) = EV(4)GO To 11

REORDERS POINTS. 202 P(3) = P(2)

P(2) = e(4)Ev(3) = EV(2)iv(2) = FV(4)GO TO 223

2 LLL = LLL 1IF (LLL ,EQ. 2) GO TO 261T = 15.60 TO 3J

C PILE 0ÉPTH SET AT MINIMUM vALLJE FOR INTEGRATIuN261 HTRUL = .001

GO TO 2888C TRUE PILE DFPTH CALCULATED

HTPuE = P(1) .5 * (0(3) - P(1))C vALuE CO M PUTED 1-OR HTPuE288 H = c)(H -MuE)

uS = HTRHEV2 = HTVAL(X9u)V 2 = AmIN1(V29Ev(2))

C VALUE COMPUTED Fop NINImuig, FROM bAYES RISSUS = 7\HINH = (XmIN)V1 = HTvgL(X9u)

C cO:ATRIuTlom FOP OPPORTUNITY LOSS CALCULATION= kpil- (v1 - v2)

C RE SETS STARTING POINT To LAST FOU NO VALUE FOR HTHuET = HTRUEIF (HTRuE.EQ0.001) T =RETURNÉND

59

5UL-i0OTI!jE SEARCH(S,XTOT)**********************************************************

CONTROL ROUTINE TO F-.;ABLE cOHPu1ATIO , '4 FOR tHEVALUE OF PILE DEPTH, ASSUING THAT MU Ai4)S%AkEo ARE THE TOUE iALi;ES SElS Ï 'RMAiEfES10 EnbLF THE 000M_E I01F(3RATION RJ3TINE TO -PP, OCELO TJJO THE GuLDFN SF(ITION sF, CH :wEN INfHE FUNCTION RR,

* **********************************************************

/S1J- -:P/AB%V:\/NO.ID. 0 Ao/5.Ei:WH9BC/PC,YHIu

COMmON /ININH/TCOmNON / cOJNT/IIIIDIMENSION LINE(135)DATA LIHE/13.5*IH*/

C SETS PAkAt'iETERS FOR SIPPSO1111 = 1ATOT = ),f = 14.H = (T/.344) ** (16 ../3.)H = 4L061(H)

C CALLS DOODLE INTEGRATION i\ID PRINTS t<ESUL1- CALL SImPSON(XToT)PRINT 2,LINE,AT0T,LINE

2 FORmA1(1H1///x9135A1///6 9*SEAr(CH FOR iIrIiu. FOR*A* EACH NU AND SIGMA S(4tJAPED*////x,* *-

*EXPECTED OPPORTUNITY LOSS =*,E2çi.e///X,133A1).-PETUR;;END

60

FUNCTION HTVAL(X,U)**********************************************************

ROUTINE THAT RETuPNS TO THE SEARCH RNOCESS. THEVALUE OF THE ObJECTIVE FUNCTION wIf ,i0or IHELIKELIHOO MULTIPLIED HY IT. COmPuftS ToF:PRO6kbILI1Y OF BRIO(iE FAILUkE dY A:4 EkOkFUNCTION IN WHIck MR. L. LANE 6AvE HIS ASSITANCL, *EVALUATES THE OBJECTIVE FuNCTIUN 46 HENTIONLI)

**********************************************************COMMON /SEAR/H9bC,PC.0 1 IN.OSCOMMO:\i /PRRR/RkDATA EP/.00001/H = AL061'AH)

C NORMALILING FACTOR FOR LOG OF FLU w= ((H-U) / SQRT(X)) / SQRT(e.)

IF (Z) 2,393Y = Abs(7)

C ERROR FuNCTION TO EVALUATE 00kmAL PkO73AHILITYCALL EHF(Y,EP.E.HH.0)

C C0i,PLITE PROBABILITY OF bRID(3E FAILOk IN ONE YEk -RRP = 05 + 5 * EIF (R - P e(T. 1.6) P,RP = 1.060 TO 4

3 CALL ERF(Z,EP.E.HH,C)RRP = .5 - * EIF (RkP .LT. 0.0) RHP = G.0

C COMPUTES PROBABILITY bR1UGE FAILURE AT LEAST k.) ,4CE IN 254 P25P = 1. - (1. -PRP) ** 25.

C EVALUATE THE OBJECTIVE FUNCTION WITHQUT THE LIKEL .i . HOUUHTVAL = (H25P * BC + PC * OS) -HETuRNEND

61

5U6ROUTIt.4E XOPT(AFA)* * * *3 * * * * 1. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * o * * * * * * * * * * * * i,‘ ** ** *1"; KES:FTS NECESSARY P.AkANETEkS TO F-_ , 6Lr. INTE0ArION ** t)VEK THi r, POSSI6LE PEA FUN FOk THL NEXT YEAk_. ** !„;ONTROL IS PA6SED TO A 0 -':E DI*NSIONAL SIP6UNS it.* UL[ FOP THE AOOVE INTEGATION. ** *o ***********************************************************

C;OMOON /AG/X (100) *X LOO ( 1 ())COHOON /SImP/XM.VAR.N.ID.GAN

C ADOS ONE NEw DATA POINT FOR NEXT YEARP',..1;INT I

1 F-,:ikFIAT(IH1//////////X.*THE FOLL041N6 PAUES OF *A*OUTPUT ARE FOR THE NEv; DATA POINTS*);,,i = N + 1

C CO:,IPUTES NEW GALHA FUNCTIOk'._ .ID = N — 1H'el = 000(N0.2)IF (MA) 21.2()921

21 6AM = GAMI(ID)00 TO 30

20 GAM = (31v ,IR(ID)30 ST = S ,:::,RT(VAP)

C SETS LIMITS FOR INTF3HATIONA = XN °. 405 * ST6 = •Xm + 4.5 * STCALL jL(A96., AREA)RETURNEND

62

SUBkouTILE RUL(A969AkEA)********** 4 **********************************************

*oNE DIALNSTOkAL SINPSONS PULE.

*******************************************************C SETS ANu LI:IITS OD INTEkvAL SIZE

DELH=03-A)/2.IF(OELH)2.1.2A!--i,EA).GO TO 7u

C CALCULATES VALUES AT EXTRENF POINTSAR1N = TAK(A) TAK(6)

C NET POINF AND LVALuAlESC=A+DELHfl = TA(C)HPAR=uELH/30*(ARIu+4.*Y1)

C CALCULATES REIJIA“JINO f-DINTS AND cHANoES CoEFFIC1ENTSOF THE PAST POI ,,1TS DURING COMPUTATIONS

30 qo 1=2915K=2**(I-1)t;K=isAREA=ARI+?.*Y1K2m1=2*K-1FL=-1.Y2=0,u0 60 M=K 9 K2H1FL=FL+2.C=A+FL*DELH/FK

6u

Y2 = y2 + TAK(C)AHEA=(AREA+40*Y2)*DELH/(3.*FK)Y1=Y14.Y2

C SPECIFIED ERROR LL-1ITEPS=At35F(AREA*1.E-i)

CHECK TO SEE IF T-)40 ITERATIONS ARE rHIN 10 PERCETIF(ABSF(AREA-RAR)-EPS)797(i,40PRAR=AREA

7u RETURNENO

FUNCTiON TAK(Y) 63*********************************************************

FuCTION EVALUATED IN 0!J•-• rYPIENSIONAL Sl'IPS0.4S kjLLFOR NEXT YEARS - POSSI6LE AT ID IS UALLL.DN .J RIAL CHI-SuAE cok;cuw,JAHTCOMPUTES NECESSARY PAPANETEPS,, SOCn AS Nr.,;4 0D VARIANCE wITH ADDITION OF IRE POINT.-

**********************************************************COMMO /1-)IP/AM9V:A.RID!6ACOM:1ON /46/X(1Lft),ALU(1)F(L) = ( SOR1(N-1.) (((N-10) * VAR)

A ** / 2.)) * ANUO ) / (SWkT(3.1415927d * N) * xpEN * (A * Z) ** ((N 1$) / 2.))INITIALIZES AND UPDATES PArfOlETEWSTOT =AN = 14°"1• = HOD(HH92)IF (H) 1,1,2XNuO = GAMI(WOr.jN = NNADEN = CiARCHN)GJ TO 3

2 xNUm = GAJARWN)H4N = NN - 1ADEN = OAMI(NN)

• SAVES OLD VALUES OF THE MEAN AND VARIANCE_3 Ad =

XV = VARALOG(,i) = Y

• CO»IPUTES NFW MEAN• = *04-1.) + ALOG(N)) / N

COPUTES NEW VARIANCEVAR = Jqt

DO 11:j i = 19N11C; VAR = VAR 4. (CALOG(I) (XL06(I) - KM)) / N

ST = S.)RT(vAP)CALL GSEARCH(ST)CALL SEARCH(ST,XTOT)YY = 10.**YPRINT lv,YY .FOR1AT(////11X,*E4 DATA POINT 4AS*,F12.2)

• RESETS PARAMETERSXVR = VAR• = 4:L3VAR = AVPROB = F(XVR)TOT = XTOT * PROBTAK = TUTRETURIArtjci

64

WORTH OF DATA STUDY RILLITO CREEK NEAR TUCSON ARIZONAMASTERS THESIS PROBLEM WILLIAM M DVORANCHIK 1970

PIER COST = 400,00BRIDGE COST = 150000,00DISCOUNT = 0.00

DATA BEGINS IN YEAR 1950

DATA POINTS 0 0

9490.0009500.0001630.0005470.0007680.0007710.0008070.0002050.0004500.0008930.000

65

We MN ,111.11 1611 LOGARITHMS

3.97726621E+003.97772361E+003021218760E+003073798733E+003088536122E+003.88705438E+003090687353E+003.31175386E+003065321P51E+003095085146E+00

are MEAN AND VARIANCE FOR SAMPLE OF SIZE 10

MEAN 3.7500272

VARIANCE = .0696379

DOUBLE INTEGRATION TO FIND THE MINIMUM PILE DEPTHOVER THE DOUBLE INTEGRATION

66

itik4.41.4 * ####** * ** * ********* ** ***** * **************** ** ****** ** **41

PILE DEPTH = 9.80 RISK CALCULATION = 1,7935188E+04

PILE DEPTH = 11,00 RISK CALCULATION = 1,1701579E+04

PILE DEPTH = 12,94 RISK CALCULATION = 8.4857637E+03

PILE DEPTH = 16.08 RISK CALCULATION = 7.4498036E+03

PILE DEPTH = 21,17 RISK CALCULATION = 8.7211287E+03

PILE DEPTH = 18,02 RISK CALCULATION = 7.7811083E+03

PILE DEPTH = 14,88 RISK CALCULATION = 7.5496138E003

PILE DEPTH = 16,82 RISK CALCULATION = 7,5486817E+03

PILE DEPTH = 15.85 RISK CALCULATION = 7,4872021E+03

i.********************************************** * * *************4

RESULTS FOR BAYES CALCULATIONS

MINIMUM 1,60832816E001

BAYES RISK 7.44980357E+03

4************************************************** ***********t

*** * ** .is.****************************************** ************A

SEARCH FOR MINIMUM FOR EACH MU AND SIGMA SQUARED

EXPECTED OPPORTUNITY LOSS 2093649538E003

4***********************ott**.atiar,****m**************it

67

WORTH OF DATA STUDY --- RILLITO CREEK NEAR TUCSON ARIZONA

MASTERS THESIS PROBLEM - WILLIAM m DVORANCHIK 1970

*****4$***44-P*.**********************************************4**********************************************************4*******************************-64**************************4

FINAL RESULTS OF STUDY

EOL OF ORIGINAL DATA = 2.936495E+03

EOL OVER ALL NEW POSSIBLE POINTS = 2,776527E+03

WORTH OF ONE MORE DATA POINT = 1.5796835E+02

41-ii, * * ********** * ********** ********************************* 4

** ***** * * **** * * ** ************ # ##** ********* * * 4-* **A. ** * ***** 4

********#4******* *******************## # ****************0-***4

REFERENCES

CDC 6000 Series Computer Systems Statistical Subroutines, Control DataCorporation, St. Paul, 1966.

CRC Handbook of Tables for Probability and Statistics, 2nd ed., TheChemical Rubber Company, Cleveland, 1968.

Davis, D. R., Department of Systems Engineering, University of Arizona,Doctoral dissertation in preparation, 1971.

Herfindahl, O., Natural Resources Infolmation for Economic Development,Johns Hopkins Press, Baltimore, 1969.

Laursen, E. M., Bridge Design and Scour Analysis, ASCE Annual Meeting- ,American Society of Civil Engineers, Louisville, Kentucky,April 1969.

Wers, B. L. and A. J. Neleher, On the Choice of Risk Levels inManagerial Decision Making, Management Sciences, Vol. 16,No. 2 (October 1969).

Raiffa, H., Decision Analysis, Addison-Wesley Publishing Co., Inc.,Reading, Massachusetts, 1968.

Raiffa, H. and R. SChlaifer, Applied Statistical Decision Theory,Graduate School of Business Administration, Harvard University,Boston, 1961.

The United States Army Corps of Engineers, Preliminary Comparisons of Analysis, Internal Working Paper, Presentation to the Bureauof Standards, Washington, D. C., 1967.

Wilde, D. J. and C. S. Beightler, Foundations of Optimization,• Prentice Hall, Englewood Cliffs, New Jersey, 1967.

68