ECONOMIC ASSESSMENT OF
SITE EXPLORATION PROGRAMS USING
STOCHASTIC DYNAMIC PROGRAMMING
Veerasak Likhitruangsilp1 and Photios G. Ioannou2
ABSTRACT
Site exploration programs are generally used to mitigate geologic uncertainty associated with
underground projects. Developing the optimal exploration program for a subsurface project is
an extremely challenging task due to the difficulty to assess its economic value in advance
and in a quantitative and systematic manner. This paper presents a methodology to evaluate
the economic value of sequential subsurface exploration programs by using stochastic
dynamic programming. The paper illustrates the application of site exploration programs as a
risk reduction strategy for underground projects. A sequential site exploration program is
structured as a stochastic dynamic programming model. Effects of test results on geologic
prediction are analyzed using Bayes’ theorem. Preposterior analysis is performed to evaluate
the economic value of the program based on the concept of value of information. A
sequential site exploration program is used to illustrate the application of the proposed
methodology.
KEY WORDS
Stochastic dynamic programming, construction risk, site exploration programs, economic
assessment, value of information.
INTRODUCTION
Geologic uncertainty is the primary source of risks in underground projects. Site exploration
programs (e.g., borehole tests) are generally adopted as a risk reduction strategy. Developing
the optimal exploration program is an extremely challenge task for all construction planners
because its economic value is difficult to assess in a quantitative and systematic manner.
In this paper, we present a new methodology to evaluate the economic values of
sequential exploration programs by using stochastic dynamic programming. The paper
illustrates the application of Bayes’ theorem to analyze effects of exploration results on
geologic prediction. Preposterior analysis is performed to assess the economic value of the
site exploration program based on the concept of value of information.
1
2
Lecturer, Department of Civil Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok
10330, Thailand, Phone +66 2 218-6475, FAX +66 2 251-7304, fcevlk@eng.chula.ac.th
Professor, Civil and Environmental Engineering Department, Rm. 2354 G.G. Brown Building, University
of Michigan, Ann Arbor, MI 48109-2125, Phone +1 734/764-3369, photios@umich.edu
SITE EXPLORATION PROGRAMS
Recognizing geologic conditions is extremely important for underground construction. This
is because subsurface conditions have a strong influence on all aspects of underground
projects, including technical feasibility, structural design, and construction planning and
estimating.
In order to appreciate geologic conditions, various site exploration programs are
generally adopted for underground construction. Typically, site investigations can be divided
into three major phases:
Phase 1: Investigation of general characteristics prior to planning,
Phase 2: Detailed geologic investigation parallel to planning and design, and
Phase 3: Geologic observation during construction
Site exploration programs performed prior to planning encompass bibliographical and
statistical surveys, morphology, petrography, stratigraphy, and hydrology of the site. The
programs also include thorough field reconnaissance and surface explorations such as aerial
photographic studies. These programs can provide general appreciation of rock type, location
and extent of major structural geology (e.g., faults), and degree of rock weathering in the
area. Even though this information is typically adequate for the feasibility study and the
preliminary design of the project, it is inadequate for detailed structural design and
construction planning. Thus, it is often necessary to additionally implement detailed
subsurface explorations.
Detailed geologic investigations are engaged parallel to project planning and design,
including geophysical surveying, exploratory drilling, test pits, in-situ testing, and laboratory
testing. These programs are used to characterize important geologic features such as rock
strength, relative density, and degree of weathering. The programs also explore the location,
extent, and condition of structural geology, as well as the location, quantity, and chemical
composition of groundwater in the vicinity of project. This information is incorporated with
the data gathered in the previous phase to predict significant geologic conditions of the site,
which are then used to prepare appropriate construction plans and construction estimates.
For example, geologic conditions of a tunneling project are typically illustrated by a
geologic profile along the tunnel alignment. The details include important rock properties,
the amount of groundwater, and the location and extent of major structural geology along the
tunnel profile. Figure 1 shows a geologic profile of a rock tunneling project developed during
the planning phase. It indicates important geologic features along the tunnel horizontal
profile such as major faults, major fractures, and lithologic contacts. It should be noted that
this geologic profile also denotes uncertainty associated with the geologic conditions (e.g.,
uncertainty in orientation and extent of a fault).
2
EXPLANATION
FAULT (DASHED WHERE
APPROXIMATELY LOCATED)
6900
6800
LITHOLOGIC CONTACT (DASHED WHERE
APPROXIMATELY LOCATED)
TUNNELING DIRECTION
MAJOR FRACTURES
6700
?
CINNAMON CREEK
BOREHOLE
Scale
200 ft
61 m
Note: 3.28 ft = 1
m
?
6300
HL-2
CROWN OF TUNNEL
AMPHITHEATER
PORTAL
1388+89
ELEV. 6150
6400
HL-2
INDICATEDS UNCERTAINTY IN
ORIENTATION OR EXTENT OF A FAULT
OF LITHOLOGIC CONTACT
Han
g
Fault ing Lak
e
6500
?
ELEVATION
(FEET)
6600
6200
-0.50% GRADE
?
?
INVERT OF TUNNEL
6100
1375 + 00
1380 + 00
1385 + 00
1390 + 00
Figure 1: Geologic Profile of a Rock Tunneling Project
(Plate 3 in Leeds, Hills and Jewett, Inc. 1981)
During construction, the geologic prediction developed in the previous phases is modified
by integrating with the prevailing ground conditions observed. In tunneling projects,
geologists are typically responsible for monitoring changes in rock type, rock conditions,
geologic structure, and hydrogeological characteristics during excavation. These field data
can be used to update existing geologic profiles (Whittaker and Frith 1990). The updated
profiles can then used for planning the remaining portions of the tunnel.
GEOLOGIC UNCERTAINTY
Since geologic conditions have a strong influence on design and construction decisions of
underground projects, knowledge about subsurface conditions directly affects the optimality
of structural design and construction plans. That is, the accuracy of site characterization
significantly contributes to the success of underground projects. In general, geologic
prediction is an extremely challenging task for all planners because it is often characterized
by a great variety of uncertainties. Geologic uncertainty is considered the primary source of
risk in underground projects (Ioannou 1987). Geologic uncertainty is referred to as imperfect
knowledge about geologic conditions at the site level. It ranges from unknown site conditions
to uncertain geologic conditions due to limited site data.
In general, subsurface conditions denoted in a geologic profile (e.g., Figure 1) are derived
from the geologists’ interpretation and projection of surface conditions such as outcrops to
the site level. This practice is the only option in several underground projects where
subsurface explorations are impossible (e.g., extremely deep underground facilities). Clearly,
the accuracy of geologic prediction by this approach mainly relies on the capability of the
geologists who interpret subsurface data and project them to the site level.
In modern underground projects, a variety of site exploration programs are available to
mitigate geologic uncertainty, including borehole tests, geophysical methods, and pilot
tunnels, leading to more accurate geologic prediction. The accuracy however is limited by
several constraints. Since site exploration programs are costly and their economic values
3
cannot be readily perceived in advance, investments on such programs are usually limited,
contributing to insufficient geologic data for design and construction decisions.
Another limitation is technological constraints; for example, subsurface explorations for
very deep tunnels may be infeasible. The interpretation of geologic conditions obtained from
geophysical methods is usually limited to a certain depth. In addition, test results from most
programs cannot perfectly characterize actual geologic conditions at the site level because of
their limited capability. In other words, test results obtained from most exploration programs
are not completely reliable. Thus, it is necessary to address this type of uncertainty while
assessing the economic value of any site exploration program as well.
ECONOMIC ASSESSMENT OF SITE EXPLORATION PROGRAMS
Geologic information obtained from site exploration programs is of importance for making
appropriate decisions during underground construction planning. In general, site exploration
programs are limitedly adopted because of their high costs. In addition, since their economic
values are difficult to quantify in advance, it is uncertain whether or not the considered
programs are worth. Several previous studies have proposed methodologies to evaluate the
economic values of site exploration programs. Extensive references and discussions can be
found in Likhitruangsilp (2003). This paper presents a new methodology for assessing the
economic values of sequential site exploration programs using the integrated concept of
applied decision analysis, Bayes’ theorem, and stochastic dynamic programming.
According to decision theory, given that the cost of obtaining new information is not
considered, the expected value of a system with new information is greater than that without
it (de Neufville 1990). For example, new geologic data provided by an exploration program
can eliminate some types of geologic uncertainty associated with the project, which may in
turn decrease the expected project cost. Thus, the economic value of an exploration program
equals to the difference between the expected project cost without the program and the
decreased project cost if the program is implemented. This difference is called the expected
value of perfect information (EVPI) if the program can provide perfect information (i.e., all
uncertainties are eliminated). It is called the expected value of sampled information (EVSI) if
the program cannot provide perfect information (i.e., uncertainty still exists). The positive
value of EVPI or EVSI means that the exploration program provides useful information
leading to a new optimal policy. Thus, an exploration program should be adopted if the cost
of such program is less than its expected value of information. In contrast, if the cost of an
exploration program is greater than its expected value of information, the program should be
rejected.
The procedure for assessing the economic value of a site exploration program consists of
three steps: (1) determine uncertainty associated with the program, (2) perform preposterior
analysis to update geologic prediction, and (3) evaluate the economic value of the program.
DETERMINE UNCERTAINTY ASSOCIATED WITH THE PROGRAM
Even though site exploration programs can be used to mitigate geologic uncertainty
associated with subsurface construction, their results are characterized by uncertainty. That
is, these results are imperfect. This uncertainty can be systematically quantified by using the
4
likelihood (reliability) function. The likelihood function of a particular exploration program
represents the likelihood of all possible test results for any given geologic parameter state.
Table 1 presents uncertainty associated with an exploration program in the form of a
likelihood (probability) matrix. For example, the first row of the matrix shows the
probabilities that this borehole test would indicate rock type ( j ) 1, 2, or 3, given that the
probability of sampled rock is type 1 (i  1) , are 0.85, 0.10, and 0.05, respectively.
Table 1: Likelihood Function of Borehole Test for Rock Type Parameter
Actual Rock Type, i
1
2
3
Rock Type Indicated by Borehole Test, j
j=1
2
3
0.85
0.10
0.05
0.15
0.70
0.15
0.15
0.05
0.80
PERFORM PREPOSTERIOR ANALYSIS TO UPDATE GEOLOGIC PREDICTION
In this step, we incorporate available geologic data with additional information obtained from
new exploration programs by using the likelihood function of the program discussed in the
previous step. This can be accomplished by performing preposterior analysis or Bayes’
updating (Benjamin and Cornell 1970). This process applies Bayes’ theorem (Bayes’ rule),
which can be expressed by the following equation:
P( B j | A) 
P( A | B j ) P( B j )
n
 P( A | B ) P( B )
j 1
j
j
where the likelihood function P ( A | B j ) is the probability that the exploration would provide
test result A given that the true state is B j . In words, the prior probability of parameter state
B j , P ( B j ) , is combined with the likelihood function of the exploration P ( A | B j ) and
normalized by the normalizing factor to yield the posterior or updated probability of state B
given that new information A is provided, P ( B j | A) .
EVALUATE THE ECONOMIC VALUE OF THE PROGRAM
For a small decision problem, it is possible to perform preposterior analysis by using a
decision tree representing the problem. The main components of this decision tree are
decision nodes of alternative exploration programs, chance nodes of test results, and costs
associated with all possible exploration decisions and test results. Examples of alternative
exploration decisions include not adopting any program, adopting a program providing
perfect information, and adopting a program providing imperfect information. Detailed
discussion about the decision tree and its solution procedure will be presented shortly.
5
Different exploration programs yield different test results. Perfect exploration programs
provide results that are exactly the same as actual conditions. That is, perfect exploration
programs can indicate actual geologic conditions. In contrast, imperfect programs yield
imperfect results, which may be different from actual geologic conditions. These uncertain
test results are considered random variables and can be characterized by likelihood functions,
as discussed above.
STOCHASTIC DYNAMIC PROGRAMMING
Stochastic dynamic programming is an efficient tool generally used for modeling and
analyzing probabilistic sequential decision processes like the economic assessment of
sequential site exploration programs. Figure 2 illustrates a symbolical representation of a
probabilistic sequential decision process.
PROBABILISTIC SEQUENTIAL DECISION PROCESS
At a particular decision stage (stage n), a decision maker observes the current state of the
system (state i). Based on this prevailing state, the decision maker chooses an action from the
set of available actions for that state. An action leads to two consequences: (1) the system
evolves to a possibly different state at the next stage (stage n  1) in accordance with the
transition probability of the system (here the system evolves from state i to state j), and
(2) the decision maker receives a reward (or a cost is incurred). Both the transition
probability and reward may depend upon the decision stage, the choice of action, and the
current and next system states (i.e., the transition). At the subsequent stage, the decision
maker encounters a similar problem, but the system may now be in a different state, and there
may be a different set of actions to choose from. As this process evolves through time, the
decision maker receives a sequence of rewards (Puterman 1994).
Probabilistic sequential decision problems can be analyzed by two main approaches:
decision trees and mathematical models. In this paper, we focus on the application of the
decision tree to the site exploration problem. Extensive discussions on the application of
mathematical models on other construction engineering and management problems can be
found in Likhitruangsilp (2004), and Likhitruangsilp and Ioannou (2004).
Figure 2: Probabilistic Sequential Decision Process
6
DECISION TREE
A decision tree can be used to diagram a small probabilistic sequential decision problem. A
tree consists of three main components: decision nodes, chance nodes, and branches. Each
decision node, symbolized by a small square, represents each decision that has to be made,
and the branches connected to the right represent all possible alternatives that can be chosen.
A chance node, denoted by a small circle, represents the random variable of decision
outcomes, the likelihood of occurrence of which can be described by a probability
distribution function.
Once the problem has been structured by a decision tree, it can be solved by the solution
procedure called rollback analysis, which begins at the end branches of the tree. Based on the
decision rule of maximizing the expected monetary value (EMV), rollback analysis consists
of two steps: (1) determine the EMV of the outcomes for each chance node, and (2) select the
alternative with the highest EMV for each decision node. The expected monetary value
(EMV) of each chance node is calculated and used as the end-branch value of the front
branch connected to that node. This procedure is performed recursively from the left to the
right of the decision tree. The calculation keeps going backward in the same manner until it
reaches the leftmost starting node.
For assessing the economic value of a site exploration program, the exploration methods
considered and their corresponding test results are presented in a decision tree, which is
analyzed by using rollback analysis. The final expected value of each alternative program is
used in computing the EVPI or EVSI of that program. By comparing such value with the cost
of the program, the decision about the program can be made. That is, the program should be
adopted if the EVPI or EVSI of the program is greater than its cost, whereas the program
should be rejected if its cost is greater than the EVPI or EVSI.
EXAMPLE APPLICATION
The following example illustrates the application of the proposed methodology to assess the
economic value of a sequential exploration program. We first investigate the case where the
contractor adopts a single test only, and then examine the case where two tests are adopted.
SINGLE TEST
A contractor needs a source of aggregates for his construction project located in a remote
area. The contractor is considering three possible alternatives: (1) develop pit, (2) buy
aggregate from suppliers, and (3) test the pit. Figure 3 shows the decision tree with all
corresponding costs, probabilities, and calculated expected values of the three alternatives.
For the first alternative, the contractor develops a pit by setting up an aggregate
processing facility, which will cost $400,000, regardless of the quantity of rock in the pit.
However, it is uncertain whether the aggregate provided by the pit would be sufficient. If the
aggregate is not sufficient (N), the contractor will have to haul as needed with an extra cost.
The second alternative is that the contractor purchases all aggregates from suppliers. This
alternative will guarantee the sufficiency of the aggregate with the total cost of $1,000,000.
7
Figure 3: Decision Tree of the Aggregate Pit Example – One Test
For the third alternative, the contractor conducts a test before making the decision to
develop a pit or to purchase the aggregate. The cost of the test (CT) is included as part of the
value of events following the test. The reliability of this test is represented by a likelihood
function. The following probabilities are estimated: the probability the aggregate is sufficient
is 2/3. If the aggregate is sufficient, the test pit will so indicate with the probability of 0.9;
and if it is not sufficient, the test will so indicate with the probability of 0.7. These
probabilities are then used in Bayes’ updating, the results from which are shown in the tree.
Solving this decision tree by performing rollback analysis yields the expected costs of the
three alternatives: Alternative 1 Develop pit = $916, 700 ; Alternative 2 Purchase all
aggregates = $1, 000, 000 ; and Alternative 3 Test option = $895, 000  CT . Thus, the test is
preferred if its cost (CT) is less than the expected value of sampled information (EVSI) of
this test, which equals to $916, 700  895, 000  $21, 700. The tree also provides the optimal
exploration policy: if the test indicates sufficient (IS), the contractor should develop a pit;
otherwise (i.e., IN), the contractor should purchase the aggregate from suppliers.
TWO TESTS
In this case, the contractor wants to conduct two successive tests for obtaining more
information, namely, Test 1 and Test 2 with the total costs of CT1 and CT2, respectively.
Figure 4 shows the decision tree of this sequential exploration program. All the probabilities
in the tree are calculated by applying Bayes’ updating. For example, the probability that the
second test indicates sufficient given that the first test so indicates, P( IS2 | IS1 ) , equals to:
8
P( IS 2 | IS1 ) 

P( IS1  IS 2 ) P( IS1  IS 2  S )  P ( IS1  IS 2  N )

P( IS1 )
P( IS1 )
P ( IS1  IS 2 | S ) P ( S )  P ( IS1  IS 2 | N ) P ( N )
P ( IS1 )
 P( IS | S )

2
P( S )   P( IS | N )  P( N )
2
P ( IS1 )
(0.9) 2 ( 2 )  (0.3) 2 ( 1 )
3
3  0.814

0.7
The other probabilities in the tree can also be calculated in the similar manner. The tree is
then solved by rollback analysis.
Figure 4: Decision Tree of the Aggregate Pit Example – Two Tests
9
As can be seen, Test 1 is worth adopting if $895,000 + |CT1| < $916,700; or $893,500 +
|CT1| + 0.3|CT2| < $916,700; that is, if |CT1| < $21,700 or |CT1| + 0.3|CT2| < $23,200. In
addition, the results also show that Test 2 is worth doing only if Test 1 indicates insufficient
(IN1) and $995,000 + |CT2| < $1,000,000; that is, if |CT2| < $5,000.
CONCLUSIONS
Subsurface exploration programs are an important risk reduction strategy for underground
projects. Decision theory can be applied to evaluate the economic values of such programs
and to establish the optimal exploration plan. Test results provide invaluable information for
both design and construction decisions. The application of decision analysis to tunnel
construction planning can be found in Likhitruangsilp (2003).
ACKNOWLEDGMENTS
The authors wish to thank Chulalongkorn University for financially supporting portions of
the work presented here. Any opinions, findings, conclusions, or recommendations in this
paper are those of the authors and do not reflect the views of Chulalongkorn University.
REFERENCES
Benjamin, J.R. and Cornell, C.A. (1970). Probability, Statistics, and Decisions for Civil
Engineers. McGraw-Hill, NY, 684 pp.
de Neufville, R. (1990). Applied System Analysis: Engineering Planning and Technology
Management. McGraw-Hill, NY, 470 pp.
Ioannou, P.G. (1987). “Geologic prediction model.” J. Constr. Engrg. and Mgmt., ASCE,
New York, NY, 113(4) 569-590.
Leeds, Hills and Jewett, Inc. (1981). Preliminary Geologic Investigation and Laboratory
Testing: Hanging Lake Tunnels, Glenwood Canyon, Colorado. Report prepared for
Department of Highways, State of Colorado, August 1981.
Likhitruangsilp, V. (2004). “Applied dynamic strategic planning for construction decisions
under uncertainty.” Proc. of the CIB W 107 2004 Symposium, November 17-19, 2004,
Bangkok, Thailand.
Likhitruangsilp, V. (2003). A Risk-Based Dynamic Decision Support System for Tunnel
Construction. Ph.D. Diss., Civ. & Envir. Engrg., U. of Michigan, Ann Arbor, MI, 503 pp.
Likhitruangsilp, V. and Ioannou, P.G. (2004). “Risk-sensitive Markov decision process for
underground construction planning and estimating.” Proc. of the Xth International Conf.
on Computing in Civil and Building Engrg., June 2-4, 2004, Weimar, Germany.
Puterman, M.L. (1994). Markov Decision Process: Discrete stochastic dynamic
programming. John Wiley & Sons, Inc., NY, 672 pp.
Whittaker, B.N. and Frith, R.C. (1990). Tunneling Design, Stability, and Construction. The
Institute of Mining and Metallurgy, London, 460 pp.
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