1.
Engineering Geology, 12 (1978) 143-161
CD Elsevier Scientific Publishing Company, Amsterdam - Printed in The Nctherlllnds
I
143
DECISION ANALYSIS APPLIED TO ROCK TUNNEL
EXPLORATION
HERBERT H. EINSTEIN, DUANE A. LABRECHE, MICHAEL J. MARKOW and
GREGORY B. BAECHER
Department of Ciuil Engineering, Massachusetts Institute of Technology, Cambridge,
Mass. 02139 (U.S.A.)
(Received November 16, 1976; revised version accepted August 10, 1977)
ABSTRACT
Einstein, H.H., Labreche, D.A., Markow, M.J. and Baecher, G.B., 1978. Decision analysis
applied to rock tunnel exploration. In: W.R. Judd (Editor), Near Surface Underground
Opening Design. Eng. Geo!., 12(1): 143-161.
Exploration planning is a process of decision making under uncertainty. The decision
if and where to explore depends on construction strategies and cost; the selection of
construction strategies del)ends on knowledge of geologic conditions which are not known
with certainty before exploration is performed. The proposed application of decision
analysis provides a relatively simple approach to the tunnel exploration problem. The
existing knowledge of geology, the possible construction strategies and their costs, the
reliability and the cost of considered exploration methods are used to establish if and
where exploration is beneficial. The resulting hierarchy of locations where exploration
is beneficial and the comparison oC expected values oC exploration Cor different exploration methods provides the basis for the selection of a particular site and method. Graphical
and simple numerical means have been created that make the proposed approach a convenient and Cast tool in the hands of the decision maker.
1. INTRODUCTION
A major problem in geotechnical engineering and particularly in
tunneling is selection of methods and locations of exploration. The goal
of exploration is to reduce uncertainty about geologic conditions, but in
planning for exploration, these uncertainties still exist. To place a value
on exploration one must place a value on "reduction of uncertainty".
This can be done only by determining the effect of uncertainty reduction
on construction cost.
Exploration planning is a classic problem of decision under uncertainty.
Due to uncertainty in the environment, several actions are possible and
decisions must be based on potential cost consequences. The application
of decision analysis to exploration is thus strongly indicated. While
statistical modeling and decision analysis have been applied to certain
problems in mining exploration, progress in applying such techniques to
2.
144
geotechnical engineering, and tunneling in particular, has been slow. The
major reason for this is lack of stochastic models for geotechnical features.
Most present approaches are based either on intuition or on simplifications,
and many contain inconsistencies which affect decisions (Lindner, 1975).
The present paper introduces a consistent use of decision analysis in
exploration for rock tunneling. A brief overview of principles of decision
analysis is given first, followed by spec:ific application to tunnel exploration.
An example concludes the paper.
2. DECISION ANALYSIS
Two topics are covered in this section: (1) principles of decision
analysis and (2) major elements of decision theory.
2.1. Principles of decision analysis
Fig.1 presents the decision-analysis cycle consisting of deterministic,
probabilistic and information phases leading to a decision (which can be a
decision to collect more information). Each of the phases consists of steps,
also shown in Fig.1. Most steps are described in detail in the application to
tunnel exploration (Section 3) and comments here are made only to clarify
the overall concept.
In the deterministic phase one enumerates courses of action and outcomes
they may lead to 1-3. Outcomes depend both on decision variables that can
be controlled by the decision maker and state variables representing the
uncontrollable environment (4). Variables and outcomes are related to each
other in a model (5). Outcomes are compared with one another through an
Qath.,
now information
Oet~,mlnl$tlc
I,
Probabilistic Ptlase
Phase
Define problem Qnd I,m,"
I"Vestloa •• on
of
I [.pre"
uncortolnly In variables by
mcons of probabilities
2 Alternative courses of action
2. P'ababll,'!oc model
3, Outcomes of eoch ollornO'ltC
3, Es'abllSh ,olal,ve value 01
p,obobllll~tl( outcomes
4.
Selec' decision and siote vanablos
5 Relale outcomes and variables
6. Method of compOtlnq rolaf. vo
vOlues of each outcome
4_ Probabilistic sensitivity analySIs
Information Phase
7 Time preference
I
8 Dominated alternotlyes ellmlnoted
2 E .. aluate various information
9, Sensl' lVlfy of outcome to vorlObles
Value of perfect information
collection schemes
Fig.I. The decision analysis cycle (after SlaCl von Holstein, 1973).
,
,
I
1
f
~
3.
145
objective function usually involving cost or time criteria (6,7). Sensitivity
studies (9) make it possible to identify most important variables.
The probabilistic phase is basically a "revision" of the deterministic phase
introducing uncertainty of the state variables. Outcomes in the probabilistic
phase are in the form of probability distributions (e.g., of cost).
A decision could be made at this point, but generally one wants to
establish whether further information gathering might be beneficial.
This is analyzed in the information phase. If information gathering is
expected to be beneficial, the results of this phase will also be the optimal
level of information gathering.
2.2. Major elements of decision theory*
Probability. Decision analysis is applied because of uncertainty about the
state of nature. Although the true state of nature is of course not random,
information about it is limited and any predictions we make are uncertain.
Experimentation can thus not be used to directly establish the level of
uncertainty, but uncertainty can be determined by our interpretation of
experimental data.
Probability has to be viewed as a subjective degree of belief which allows
us to reflect all available information, experience and judgement (for further
details on this approach see, e.g., Baecher, 1972).
Decision criteria. To judge the relative value of alternative outcomes a
decision criterion must be chosen that is consistent with the situation.
In our case this is a situation of risk (outcome not known with certainty,
but the probabilities of alternative outcomes can be evaluated since we use
a subjectivist approach). The assessment of risk and thus the selection of
decision criteria depends on the impacts of the decision that are deemed
most important by the decision maker. The most general measure of such
impacts are utilities which can express the attitude of the decision maker
toward deviations in cost, time, quality, etc. In our case the measure of
impact of an outcome and thus the decision criterion, is simply the expected
cost.
~
Preposterior analysis. This is a key element in the information phase. The
consequences of potential future actions (collecting new information) are
assessed before the action is taken. Preposterior analysis makes use of
Bayes' theorem to perform the so-called Bayes Updating.
Thomas Bayes formulated his theorem in a paper published in 1763
(see Bayes, 1958). Benjamin and Cornell (1970) give a simple two-step
*For an introductory level discussion of decision analysis sec Raiffa (1968).
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derivation of this theorem which can be written as:
P[BjIA]
=
nP[AIBilP[Bj]
~P[A IBj ] P[Bj
]
j;l
Or in words:
Posterior probability]
[Likelihood of the] [priOr
~ [NOrmaliZing]
of Bj given the new
= new.information, x probability x factor
[
information, A
A, gIven Bj
of Bj
The likelihood function P[A IBj ] is the probability that the particular
observation A (or "data A") would be made in exploration, given that the
true state of nature was Bj •
In Bayesian updating the prior probabilities of the state variables P[Bj ] are
combined with the likelihood function of the new information P[A IBj ] to
determine posterior - or "updated" - probabilities. These in turn are used to
perform a decision analysis by calculating probabilities of the outcomes of
each exploration alternative occurring. The value of the new information
can be determined as the difference between expected values of total
construction cost and time before and after gathering the new information.
A convenient means to conduct the preposterior analysis is the decision
tree whose use will be shown in the application of the decision analysis to
tunnel exploration (see also Section 4).
With this introduction to decision analysis, it is now possible to present
its application to tunnel exploration.
3. APPLICATION OF DECISION ANALYSIS TO TUNNEL EXPLORATION
The decision analysis phases and steps of Fig.l are now (Table I) related to
the tunnel exploration problem. A few comments will be made regarding those
steps.
The alternatives "exploration" or "no exploration" (step 2) can be defined
for any stage of a geotechnical study, either the very first exploration or
additional exploration at later stages. The "exploration" alternative includes
many exploration strategies which are combinations of methods, locations
and numbers of explorations. An exploration method is characterized by its
cost and its reliability. The reliability is the probability that the exploration
results indicate the true conditions and is represented by the likelihood
function in Bayes' theorem.
The value of information (step 3) is the difference between expected
construction cost without exploration and expected construction cost with
the particular exploration alternative. The goal of exploration usually is to
reduce the expected construction cost. However, exploration involves some
cost also. The objective of the decision analysis is thus to minimize
~
5.
147
TABLE I
Decision analysis for tunnel exploration; steps in the analysis of the exploration decision
Step Phase
Exploration for tunnels
1
Deterministic
Define the decision problem
2
Identify the alternatives
Exploration
No exploration
3
Outcomes
Value of information; construction costs and
exploration sites are intermediate results
4
Decision and state variables
Exploration cost
Exploration reliability
Construction method costs
Geology
5
Relationships between variables
and outcomes
Effect of geology and exploration on
expected construclion cost; decision tree
6
Value (of outcomes)
Expected cost
1
Probabilistic
Encode uncertainty in state variables
2
Probabilistic model
Effect of geology and exploration on
expected construction cost; decision tree
3
Choose among distributions
e.g., mean of cost·time scattergrams
4
Probabilistic sensitivity
analysis
Critical ranges of probabilities, exploration
reliability, and constructions costs
established
1
Information
Value of perfect information
2
Best information gathering scheme
Determine if exploration is beneficial; if it is,
what is the optimal exploration program
Prior probabilities of geologic states
Probabilistic sensitivity analysis
Exploration method and configuration
(geometry along tunnel)
exploration cost plus expected construction cost, or in other words, to
establish the maximum value that one is willing to pay for exploration.
The decision variables are the exploration methods which can be
described by exploration cost and reliability and the construction methods
which are described by their costs.
The state variables are the geologic conditions affecting tunnel construction
such as jointing, water inflow, major defects. In this paper we are using
a simplified description of geologic conditions with the three states good.
fair and poor.
The establishment of relations between variables and outcomes is the
major problem that needs to be solved in the decision analysis; it will be
discussed in detail in Section 4. At this point, it may suffice to say that
6.
148
geologic conditions, exploration and construction costs have to be
related to each other. The value of outcomes in the present case is taken
to be expected cost of construction, in other words a uniattribute linear
utility function over monetary cost.
In the probabilistic phase, degree of belief probabilities are assigned
to the geologic states. (See Section 4 and Einstein and Vick, 1974). The
probabilistic model should make use of the deterministic relations and
in addition, by introducing uncertainties in the form of subjective
probabilities, produce distributions of outcomes instead of single values.
One means for relating geology to construction cost in a probabilistic
manner is the Tunnel Cost Model (Moavenzadeh and Markow, 1976).
A sensitivity analysis is performed to evaluate changes in the "best"
decisions and changes in predicted costs that result from variation in the
input parameters. In this way, the "sensitivity" of optimal exploration
strategies to probabilities of geologic conditions and estimates of
construction sequences can be determined. If the optimal strategies
are insensitive to minor fluctuations in the variables, then we say that
the decisions are "robust", and we have more confidence in them (Some
details on the sensitivity analysis will be given in Section 4.)
In the information phase, the expected value of perfect information
(EVPI) is calculated which eliminates those sections of the tunnel in which
even perfect information (e.g., knowing the true geologic conditions
precisely) would not be cost effective. Then, in the remaining sections,
the expected value of alternative imperfect exploration schemes is
evaluated.
We have now established the general procedure with which the decision
analysis can be applied to the tunnel exploration. To make the procedure
practically useable some details of the decision analysis need to be
described.
4. SOlIE DETAILS OF THE DECISION ANALYSIS
Facets of the decision analysis approach that need further discussion
specifically related to tunneling are the relationships among input variables
and outcome, the comparison among outcomes, and the sensitivity
analyses. A tool for relating variables and outcomes and for comparing
different outcomes is the decision tree:
4.1. Decision tree
The relationship among variables can be organized in a tree structure
(Fig.2) for easier manipulation.
The expected cost of the no exploration case is computed by: (1)
multiplying the cost of any of the construction strategies in a particular
geology by the originally estimated subjective probability of that
geology; (2) summing these "expected costs" for each construction
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149
EXPLORATION
METHOD
EXPLORATION
RESULTS
CONSTRUCTION
METHOD
TUNNEL
GEOLOGY COST (5)
lOO
640
51
...-E--~~ ~OO
1500
t
Prior
PrObablilf les
lOO
20'0
~OO
51
52
53
500
600
600
51
52
53
t
Probabllilies
of Expioralion
Results
I
52
300
~
.2
.6
_
~
515
51
515
.34
52
53
51
52
53
543
589
603
1
Posterior
ProbabilitIes
J
300
600
1000
SI
52
S3
Fig.2. Exploration decision tree.
strategy (e.g., 81 = 640); and (3) selecting the construction strategy with
minimum expected cost (83 or 82 = 620).
The expected cost for the exploration cases are computed similarly by:
(1) calculating the posterior probability of each geologic state conditioned
on each possible result of exploration (e.g., if the exploration program
indicates "fair" geologic conditions, the probabilities of "poor", "fair",
and "good" conditions might be 0.09,0.78, and 0.13, respectively);
(2) determining the expected cost (Le., probability times cost of any of
the construction strategies in the particular geology) for each exploration
result (analogous to step 1 above: 300 x 0.09, and so on, and these
summed); (3) selecting the minimum expected cost construction strategy
for each exploration result; (4) finally, weighing each minimum cost
strategy by the probability of the corresponding exploration result and
summing (515 x 0.34 + 603 x 0.46 + 680 x 0.20 = 588). Adding the cost
of the particular type of exploration to this sum yields the expected
total cost of that exploration strategy (588 plus cost of exploration).
8.
150
The expected value of exploration (or expected value of sample
information EVSI) is the difference between the expected cost of the best
strategy (step 4) and the expected cost of the "no-exploration" case
(Le., 32 minus cost of exploration).
It should be kept in mind that the analysis expressing these relations has
to be performed for each location of possible exploration.
4.2. Sensitivity analysis
In the sensitivity analysis the above described relation (tree) between
variables and outcome is used. All the variables are varied and the EVSI for
each combination of variables is determined. The range of variation of the
construction costs is usually estimated in a preliminary analysis (see also
Sections 5.1 and 5.2), the probabilities of geologic conditions vary between
completely reliable to completely ambiguous.
The exploration reliability is expressed in the form of a reliability matrix,
a matrix of likelihood functions as shown in Table II. The likelihoods or
reliabilities are the result of subjective assessment of the performance of an
exploration method in a certain geologic condition (e.g., the method
described in Table II has an 0.5 ... reliability showing "poor" conditions
if the real conditions are "poor"). In assessing the reliability one has to
keep in mind that we consider entire segments; the reliability of a single
boring may thus decrease as the segment length increases. As mentioned
before, the reliabilities vary between completely reliable (diagonal
probabilities = 1, rest = 0) and completely ambiguous (all probabilities
equal1/n, where n = number of geologic states).
The construction costs of a certain construction strategy in certain
geologic conditions can also be represented in matrix form (Table III).
Assuming that the strategies are ideal for the geologies along the diagonal,
one can form a so-called penalty matrix with O-penalties along the
diagonal (upper part of Table IV). If a strategy is the best for all geologic
conditions, a row of O-penalties would occur; for the numerical example
TABLE II
Reliability matrix·
Exploration result
EF
Ep
EG
G
0.2
0.2
F
0.3
0.6
0.1
p
Geologic states
0.6
I 0.2
0.3
0.5
• The numbers in the reliability matrix are the P[ true geologic statelexploration result].
For example, the probability of poor geology at a particular site given that the exploration
showed poor geology is (circled): P[PIEpl " 0.5.
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151
TABLE III
Construction cost matrix
Geology
G
F
81
Construction strategy
CIG
G
P
ClI:
C2G
C2F
C2P
83
C3G
C3F
P
81
e.g.
C3P
200
500
1500
82
300
600
1000
83
ClP
82
F
500
600
800
in Table IV, strategy 81 is the best for "good" and "fair" geology,
stratef,'Y 83 for "poor" geology.
The contour diagram of Fig.3 is a convenient graphical means for
carrying out the calculations of the decision tree when there are only
three geologic states. The prior probabilities of each state (e.g., for "good",
"fair", "poor") are plotted along the three axes, and the expected value
of exploration is plotted as contours over the triangular grid. One such plot
describes thus the expected value of exploration for a specified penalty
matrix and reliability matrix.
The value of perfect information (as introduced in Section 4.1) for a
given set of prior probabilities equals the minimum expected penalty cost
over possible construction strategies minus the penalty associated with
perfect knowledge of the geologic state, which is zero. Calling this the
"expected value of perfect information" (EVPI),
where PG,F,p = prior probabilities (of good (G), fair (F), poor (P)
conditions); PSG = penalty cost of strategy S, in good (G) geologic
conditions.
TABLE IV
Penalty matrix
G
81
F
P
G
F
P
ClG-ClG
CIF-C 2F
CIP- C 3P
0
Pu,
PIP
P2G
0
P2P
PaG
P3~'
0
82
C2G-CIG
C2~·-C2F
CaG-ClG
C3F- C 2F
~
C2P- C 3P
83
C3P- C 3P
e.g., from Table III:
P
F
G
81
0
0
700
82
100
100
200
83
300
100
0
10.
152
Good
10
10
2 (90)
,.~
20" (80)
30" (70)
4 0 (60l
50
50
-t
40
~
30
~~'"
20 •
10
Fo"
10
/
10
20
(901 (80)
30 40
(70) (60)
50
50
(GO)
40
(70) (80l (90l
30
20
10
Poet
10
Penalty Malti.,
F
G
51
P'G , 0
p,.
52
Pz.
PZF' :
53
P)Q '100
'100
P
'100
p'p '00
0
P2 • '00
p). '100
p]P
:
0
Fig.3. Contour diagram and corresponding penalty matrix (for perfect information, i.e.,
100% reliability of exploration).
Fig.3 is the contour diagram for the penalty matrix shown in the lower
part of this figure.
The expected value of sample information (EVSl) is less than the EVPI
by some factor which depends on the reliability of the exploration
technique. For 100% reliable technique, this factor is 1.0, and diminishes
to zero for a uniform reliability matrix (Le., one for which all R jj = lIn,
where n is the number of geologic states). The EVS/ can be calculated by:
(1) Determining the updated probabilities of the geological state
conditioned on each possible exploration outcome (Bayes' theorem).
(2) Determining the minimum expected penalty for each updated
probability distribution.
(3) Averaging over the probability of exploration yielding each possible
result (Le., taking the weighted sum).
This procedure can be performed with a simple computer program and
the contour plot can be correspondingly restructured. FigA shows the
contour plot for the same penalty matrix as in Fig.3, but for a 90%
reliability of exploration as expressed by the reliability matrix in FigA.
(It should be noted that 90% reliability refers to the diagonal, Le., an
exploration result "good" has 90% probability of representing the true
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153
Good
10
ConlOUt 40 is Ih. Conlour Corresponding 10 Explcwation Cosl.
5eQmtnls which Piol in th. U'lshod.d Zone are therefor.
Polenliolly Voluobl. Explcrotion SitlS.
R.I=ility MOlflx:
Ep
G
09
01
0
F
0
0.9
01
P
0
01
09
Fig.4. Contour diagram for reduced reliability of exploration and with exploration cost
contour (same penalty matrix as for Fig.3, but 90% reliability as expressed by reliability
matrix).
geologic condition "good". Thus reliability matrices with different
non-diagonal members can also represent 90% reliability of exploration.)
One of the major uses of the sensitivity analyses and the contour plots
is in locating the sites where exploration is beneficial: the exploration
value contour corresponding to the exploration cost is drawn as in the
contour diagram in FigA. The geologic probabilities of the particular
tunnel segment can then be plotted. If they fall inside the exploration
cost contour, exploration is beneficial.
5. DESCRIPTION 01<' THE DECISION ANALYSIS PROCEDURE FOR TUNNEL
EXPLORATION
Up to this point, the theoretical development of the exploration decision
problem has been described as well as important tools which will be used
in the analysis. Based on this step by step formulation of the problem,
a procedure for analysis was developed and will now be described.
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154
5.1. The problem of cyclic interdependence of construction and exploration
and its solution
One of the advantages of our approach is the segmentation of the tunnel
geology. Geologic conditions are assumed to be constant over a certain
length of tunnel; i.e., the geologic variables and their probabilities of
occurrence are constant (which of the geologic conditions actually will
occur is however uncertain as expressed by the probability). The
segmentation corresponds well to construction procedures that are modified
in discrete steps; it has the additional advantage that segments in which the
exploration would be beneficial can be identified and ranked. The discrete
(segmented) character of the construction and exploration procedures has,
however, some disadvantages that need to be overcome.
The construction cost within any segment cannot be assumed to be
simply the cost of driving the tunnel through the type of rock which exists
in that segment. In general, the construction cost is dependent not only
on the geologic conditions in a given segment, but also on the conditions
in previous, and possibly later, sections of the tunnel. (e.g., the switching
from full face excavation to heading and benching and back to full
excavation in crossing a shear zone may result in a greater cost than if
heading and benching had been used throughout. This is due to costs
related to switching which may more than offset the greater cost of
heading and benching). Also, since the improvement of knowledge of
geologic conditions by exploration will influence one's choice of
construction method, it is possible that exploration in one segment can
affect the entire tunnel. Therefore, it is not sufficient to analyze the value
of exploration on a simple segment by segment basis; it must be done on an
entire tunnel basis.
The decision if and where exploration is necessary depends on the
planned construction strategy and the projected construction cost. The
projected construction cost depends on the knowledge of geologic
conditions which means that construction cost in tum depends on
exploration. The exploration decision problem is thus cyclically intertwined
with the tunnel construction cost and construction strategy. In order to
break into the cycle, an approximation had to be made:
Each segment is analyzed individually, but is always considered as a part
of an entire tunnel profile: within the segment being analyzed, the
geologic state successively takes on each possible value, while the remainder
of the tunnel takes on the most likely value. E.g., in analyzing segment 3 in
Fig.5, the profiles of interest are (symbolically): (1) CBACA; (2) CBBCA;
(3) CBCCA. The different geologies within the analyzed segment produce
an effect on total construction cost from which in tum the value of
exploration in that segment can be determined (with the tree or the contour
plot as explained in Section 4).
13.
155
3
4
5
Prior Geoloole Probabilities of Condition;
A
0
.2
.6
0
7
B
.3
.7
3
.1
2
C
.7
I
,I
9
I
Fig.5. Segmented tunnel with prior probabilities of geologic conditions A.
n, C.
5.2. Construction cost
After establishing the possible profiles by in tum applying the procedure
described above to each segment, the cost of construction for each profile
will be estimated. This can be done easily if records of past performance for
typical geologic conditions and construction strategies are available, or by
using the Tunnel Cost Model (see Moavenzadeh et a1., 1976) or any other
cost estimation tool. Next a construction cost matrix is created as shown
in Table III, but now relating construction cost to entire geologic profiles
instead of a single geological state. The construction cost matrix is then
transformed into a penalty matrix (analogous to Table IV).
5.3. Exploration value
Once the penalty matrix is formed, the contour diagram is drawn as
described in Section 4.2. The exploration-cost contour is also drawn to
define the explore and no-explore regions. By plotting the prior
probabilities for the segment being analyzed, a decision can be made with
respect to exploration or no-exploration, and for those segments in which
exploration is beneficial, an expected value of exploration can be
determined. For example, if a segment had construction and exploration
costs as characterized by the plot in Figo4, and prior probabilities of
0.3-004-0.3 (good-fair-poor), it would have an expected value of
exploration of 50. This value can be used to establish a hierarchy for
exploration in individual segments.
This procedure is usually satisfactory since once the decision is made to
explore in a particular segment, the exploration will be performed and the
results used to update our knowledge on geologic states. Only then will a
decision be made where and how to explore further. There are instances,
however, where the decision will involve a combination of segments that are
to be explored and even differerit exploration methods in different segments.
The analysis described in this paper does not deal with these so-called
exploration strategies.
14.
156
5.4 Other aspects of the decision analysis procedure for tunnel exploration
In this section, we shall shortly discuss aspects of the decision analysis
procedure that have not been commented upon either because they are
neglected in the simplified approach presented here or because the clarity
of the description would have been reduced.
The variables that we consider (exploration cost, exploration reliability,
construction cost and geologic conditions) are only four of the possible six
variables. The other two are length and location of the considered segments.
The possible importance of these two variables has been indirectly mentioned
in the discussion on cyclic interdependence of construction and exploration.
The effect of these variables is not fully understood, but seems to depend on
the cost differences between different methods and on the cost and number
of construction method changes.
As mentioned before, the analysis of exploration strategies - combinations
of exploration in different segments and with different methods has not been
treated here because of the frequent step-by-step character of exploration
planning which makes consideration of exploration strategies unnecessary.
A last clarifying point concerns the number of geologic states which we,
in this paper, have limited to three. This is the maximum that can be handled
graphically. By using numerical methods, any number of geologic states that
are considered important can be handled. It should be noted, however, that
in the exploration planning stage, simplifications and thus three or even only
two geologic states will usually be satisfactory.
5.5 Summary of the decision analysis procedure
The decision analysis procedure can thus be summarized as follows:
(1) Determination of possible geologic states, segmentation and assignment
of (subjective) prior probabilities to the geologic states.
(2) Selection of possible construction strategies and of their costs
(including transition costs) in the various geologic states.
(3) Determination of construction costs f'Jr all possible geologic profiles
based on (1) and (2) above.
(4) Analysis for each segment consisting of:
(a) Variation of geologic states in the considered segment, with the
remainder of the geologic profiles having the most likely geologic states.
This results in a set of geologic profiles.
(b) Determination of construction cost and penalty matrices for the set of
geologic profiles determined in (a).
(c) Determination of the values of exploration by drawing the contour
plots as described in section 4.2.
(d) Drawing of the exploration cost contour and plotting of prior
probabilities. This yields an exploration value, which indicates if exploration
is beneficial and provides a means for ranking different segments with regard
to benefit of exploration.
15.
I,
157
(5) Evaluation of various exploration methods and establishment of
hierarchies for these exploration methods.
This procedure is now illustrated in an example.
6. APPLICATION OF DECISION ANALYSIS
In order to illustrate the procedure which has been described in Section 5,
a 10,OOO-ft. example tunnel, whose characteristics are presented in Fig.6, will
be analyzed (Sections 6.1-6.5) to determine the location of potentially
valuable exploration sites.
6.1. Geologic states, segmentation and prior probabilities
Based on preliminary exploration data, the two geologic conditions
"good" and "poor" have been determined. Also, the preliminary exploration
led to a "segmentation" of the tunnel, and to prior probabilities indicating
which of the two geologic conditions might exist in each segment.
6.2. Selection of construction strategies
Three construction strategies were selected for evaluation:
(a) Tunnel boring machine (TBM) throughout the tunnel.
(b) Drill and blast (D-B) throughout the tunnel.
(c) A combination of TBM and D-B in which D-B is used in poor rock and
TBM in good rock with the following qualifications: (1) if D-B is used in
segments 1 and 2, the tunnel should be completed with D-B; (2) a switch
from D-B to TBM should be made only between segments 1 and 2, i.e., if
poor rock occurs in segment 1 and good rock in segment 2.
Costs are based on the assumption that the original construction plans
called for TBM. Any switch of method will involve a transition cost.
The estimated construction costs for TBM and D-B as well as the method
change or transition costs are given in Table V.
6.3. Profile construction costs
The total construction cost for each of the possible profiles is determined
using the costs listed below. The geology of each profile, which is represented
SEGMENT
LENGTH
4000
0
1000
:3
4
2000
3000
(tt)
P"or Probob.1o lies
GEOLOGY
rl
Poor
.8
.2
~
6
4
:3
I
7
Fig.5. Example tu nnel for exploration evaluation.
I
16.
158
TABLE V
Estimated unit construction costs and transition costs
Estimated unit construction costs ($/ft.)
Geology
Method
good
500
800
TBM
D-B
poor
3000
1900
Transition costs ($)
TBM to D·B
near portal (segment 1)
in other segments
2nd method change
D·B to TBM
720,000
750,000
200,000
20,000
by a certain percentage (probability) of good and poor conditions, and its
associated cost are plotted in Fig.7. From Fig.7, it can be seen that each
strategy has a range of geologic conditions (profiles) over which it is the best
construction method.
6.4. Analysis Of each segment
Each segment is analyzed in decision tree fashion. In this example, we use
the contour diagram method. The calculations will be shown for segment 1
only.
(a) Geologic profiles obtained by varying the geology in segment 1 and
keeping the most likely geologies in the other segments: (1) GPGP; (2) PPGP.
(b) Construction cost and penalty matrices for each set of profiles (Tables
VI and VII).
(c) Determination of the expected value of exploration. The reliability
matrix for the exploration method is shown in Table VIII.
With the prior probabilities from Fig.6 and the construction penalties
listed in Table VII, it is now possible to compute the expected penalties for
the no exploration case:
Comb 2:
D-B:
(0.8)(0) + (0.2)(0.72) = 0.144
(0.8)(0.45) + (0.2)(0) = 0.36
Comb 2 has the minimum expected penalty and will thus be the selected
strategy.
For the determination of the expected penalties of the exploration case,
one computes first the posterior probabilities using the prior probabilities
and the exploration reliability listed in Table VIlI:
P'[GIG]
P'[GIP]
= 0.96
= 0.48
P'[PIG] = 0.04
P'[PIP] = 0.52
J
17.
159
30r
~
=
o
o
25
--TBM
- - O-B
COMB 2
· _.I
:
.
I
o
"
u
:>
I
·---=7"""5........~25'•
t
t
I
5-5
I
25-75
0-1
Poor
%
Tun-no I
Good· POOl
Fig. 7. Construction costs for different geologic proriles and different construction
strategies.
The computation of the posterior probabilities also yields the probabilities
of exploration (the probability that the exploration result is "good" or
"poor"):
P[G) = 0.67
P[P)
= 0.33
The expected penalties are then computed for the possible exploration
results:
Exploration "good"
Comb 2:
(0.96)(0) + (0.04)(0.72)
D-H:
(0.96)(0.45) + (0.04)(0)
4
=
=
0.029
0.431
Exploration "poor"
Comb 2:
(0.48)(0) + (0.52)(0.72) = 0.374
D-H:
(0.48)(0.45) + (0.52)(0) = 0.216
Combination 2 yields the minimum expected penalty if the exploration result
is good, D-H if the exploration result is poor. The minimum expected
penalties are now combined by taking their weighted sum (weighted by the
probability of the exploration result): 0.67 x 0.029 + 0.33 x 0.216 = 0.090.
The expected value of sample information is the difference between the
18.
160
TABLg VI
Construction cost matrix
(millions of $)
J
Geologic profile
Strategy
1
2
TBII
15.0
25.0
D·B
12.4
16.8
COMB 2
11.95
17.52
TABLE VII
TABL!'~
Penalty matrix
(millions of $)
Reliability matrix
VIII
Exploration result
Geologic profile
G
Strategy
I).B
o
0.72
0.45
COMB 2
P
G
0.8
0.2
P
0.15
0.85
2
1
o
Geologic
state
expected costs (penalties) of the no exploration and the exploration case,
the latter including the cost of performing the exploration:
EVS/
= $144,000 =
($90,000 + cost of exploration)
54,000 - cost of exploration
Exploration is thus beneficial in segment 1 if the cost of exploration is less
than $54,000.
6.5. All segments will have to be evaluated in the same manner to establish a
hierarchy
7. CONCLUSIONS
Exploration for tunneling should lead to a reduction of construction costs.
Since exploration itself involves a cost, the goal of exploration planning is to
minimize the total cost of construction plus exploration. However, the
decision if and where to explore depends on construction strategies and cost
whose selection in turn depends on our knowledge of the geologic conditions.
The exploration planning problem is thus a problem of decision under
uncertainty with a cyclic nature as a complicating feature.
The proposed application of decision analysis provides a relatively simple
approach to the tunnel exploration problem. The existing knowledge of
geology, the possible construction strategies and their costs, the reliability
~
19.
161
and the cost of the considered exploration methods are used to establish if
and where exploration is beneficial. The resulting hierarchy of locations
where exploration is beneficial and the comparison of expected values of
exploration for different exploration methods provides the basis for the
selection of a particular site and method. Graphical and simple numerical
means have been created that make the proposed approach a convenient and
fast tool in the hands of the decision maker. The approach has been
purposely kept simple and is based on several approximations, a fact which
has to be considered if the user wants to develop it further.
EPILOGUE
The application of decision analysis to geotechnical exploration and
particularly to tunnel exploration is new and untested except for a few
example cases. It is just the purpose of this paper and particularly of the
simplified approach presented in it to induce the practitioner to use decision
analysis in tunnel exploration.
The soundness and applicability of decision analysis have been proven
extensively in other areas where decisions under uncertainty have to be
taken, notably in business administration. Since the managers usually decide
on the expenditures for exploration and since they base their decisions to a
large extent on decision analysis, it seems opportune that engineers and
geologists start talking in the same language.
ACKNOWLEDGEMENTS
The research on which this paper is based has been conducted in
connection with the development of the Tunnel Cost Model (TCM)
sponsored by NSF-RANN. Prof. F. Moavenzadeh is the principal
investigator of the TCM project.
REFERENCES
Baecher, G.B., 1972. Site Exploration: A Probabilistic Approach. Ph.D. Thesis,
Massachusetts Institute of TechnoloftY, Cambridge, Mass.• 515 pp.
Bayes, T. 1958. Essay toward solving a problem in the doctrine of Chaines. Biometrika,
45: 293-315 (reproduction).
Benjamin. J.R. and Cornell, C.A.• 1970. Probability. Statistics and Decision for Civil
Engineers. McGraw-Hili, New York, N.Y., 684 pp.
Einstein, H.H. and Vick, S.G., 1974. Geoloftical model for a tunnel cost model. Proe.
Rapid Excavation and Tunneling Conf., 2nd,lI: 1701-1720.
Lindner, E.N., 1975. Exploration: Its Evaluation in Hard Rock Tunneling. jIS Thesis,
Massachusetts Institute of Technology, Cambridge, Mass., 210 pp.
Moavenzadeh, F. and Markow, M.J., 1976. Simulation model for tunnel construction
costs. J. Contr. Div., Am. Soc. Civ. Eng., COl: 51-66.
Raiffa, H., 1968. Decision Analysis. Addison-Wesley. Reading, Mass., 309 pp.
StaiB von Holstein, C.S., 1973. In: Readings in Decision Analysis. Stanford Res. Inst.,
Stanford. Calif., pp.97-216.
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