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Decision Analysis Applied to Tunnel Exploration Planning. I: Principles and Case Study

Karim S. Karam1; Jad S. Karam2; and Herbert H. Einstein3

Abstract: Exploration planning is a process of decision making under uncertainty. This paper, Part I of two dealing with the exploration problem, briefly reviews the decision analytical procedure for tunnel exploration, and provides practical techniques to do so. Specifically, this paper presents an approach by which the effect of additional exploration can be assessed before actually committing to exploration, through a process of so-called virtual exploration. This is practically done with an easy-to-use software package developed in Visual Basic for Applications in Microsoft Excel, called the Decision Aids for Tunnel Exploration. Part II will describe the uncertainties that enter into the tunnel exploration decision procedure, and will provide practical techniques to asses the effects of these uncertainties on the explo- ration decision making.

DOI: 10.1061/�ASCE�0733-9364�2007�133:5�344�

CE Database subject headings: Tunnels; Decision making; Uncertainty principles; Case reports.

Introduction

Tunnel exploration planning is a process of decision making under uncertainty. Fig. 1 is a graphical representation of the de- cision process similar to that proposed by Raiffa and Schlaifer �1964� and Stael von Holstein �1974�, but adapted to engineering where one determines parameters, includes them in engineering models, and makes decisions based on the model results.

Einstein et al. �1978� discuss the application of decision theory to exploration planning. Although the principles and methodolo- gies have existed for some time, not much has been done to make this theory practically usable. This paper attempts to change this through the development of a simple computer program and an application to a case history. Also, the consideration of switchover costs between construction methods has not been included in the previous analysis, and this is done here. �In addition, the investi- gation of the effects of exploration uncertainty, which is discussed in Part II, has been done to a limited extent only so far.� Hence this paper discusses, after a brief review of the principles, the practical techniques that have been incorporated into a computer program, Decision Aids for Tunnel Exploration �DATE�. DATE is developed in Visual Basic for Applications in Microsoft Excel,

which is widely available, and is easy to use. DATE allows one to asses the consequences of potential future actions �collecting new information through “virtual” exploration� prior to actually per- forming �or not� the exploration. This will be demonstrated in a case study by devising an optimal exploration plan for the Su- cheon Tunnel in Korea.

Background

In the following, the basic concepts of the application of decision theory to tunnel exploration are briefly reviewed. Emphasis is placed on the probabilistic and information phases �shaded boxes in Fig. 1�.

Probabilistic „Model… Phase

The probabilistic �model� phase in Fig. 1 is related to the tunnel exploration problem in Table 1. In the probabilistic phase, one estimates the tunnel cost based on the uncertain geology as ex- plained below.

1Former Research Assistant, Dept. of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139.

2Former Research Assistant, Dept. of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139.

3Professor of Civil and Environmental Engineering, Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, Cambridge, MA 02139 �corresponding author�. E-mail: [email protected]

Note. Discussion open until October 1, 2007. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and pos- sible publication on January 4, 2006; approved on October 19, 2006. This paper is part of the Journal of Construction Engineering and Manage- ment, Vol. 133, No. 5, May 1, 2007. ©ASCE, ISSN 0733-9364/2007/5- 344–353/$25.00. Fig. 1. The decision cycle

344 / JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / MAY 2007

Step 1: Uncertainty in State Variables In Step 1 of the probabilistic �model� phase, uncertainty in the geology is expressed by assigning prior probabilities �or distribu- tions� to geologic states. Note that the term “geologic states” is purposely used instead of geology, because other factors that af- fect the tunneling process such as hydrologic conditions can be included.

Step 2: Probabilistic Model In Step 2, a probabilistic model, in the form of a decision tree, is used to determine the effects of uncertain geologic states on ex- pected construction costs. Steps 1 and 2 are discussed in this paper.

Steps 3 and 4: Choose among Distributions and Probabilistic Sensitivity Analysis In Step 3, the uncertainties in state variables are expressed using probability distributions. This also includes the uncertainties as- sociated with assigning prior probabilities to geologic states since the prior probabilities are themselves uncertain. In Step 4 the effects of these uncertainties on expected construction costs are evaluated. Steps 3 and 4 are addressed in Part II.

To illustrate Steps 1 and 2 of the probabilistic �model� phase, consider the topography and tunnel alignment shown in Fig. 2.

The tunnel is divided into sections, three in this example, which are assumed to be independent. In each section, different geologic states may be encountered and, consequently, different construction strategies might be used. Table 2 shows the prior probabilities of geologic states �state variables� in each tunnel section. The construction strategies �decision variables� and asso- ciated costs per unit length are shown in Table 3. Note that con- struction strategies do not necessarily imply construction meth- ods; strategies can refer to the same construction method if, for example, different tunnel supports are used.

The total expected cost of constructing the tunnel is comprised of two main components, namely, �1� the expected cost of con- structing each tunnel section; and �2� the expected cost of chang- ing �where relevant� construction strategies between adjacent sec- tions �switchover costs�.

These components are discussed below.

Cost of Constructing Each Tunnel Section The cost of constructing a tunnel section is obtained by consider- ing the section independently of others. A probabilistic model �decision tree� is constructed for each section. Fig. 3, for example, shows the decision tree for Section 2 of the tunnel.

Chance nodes show the expected costs for a given construction strategy, and are computed from

E�C�S j� = � i=1

PiGCji �1�

where iG represents geologic state, n�total number of geologic states, S j�construction strategy, PiG��prior� probability of geo- logic state I, Cji�cost of construction strategy j in geologic state iG.

In Fig. 2, Eq. �1� results in

E�C�S1� = �0.4 � $ 1,890� + �0.6 � $ 1,935� = $ 1,917

E�C�S2� = �0.4 � $ 1,125� + �0.6 � $ 2,520� = $ 1,962

At the decision node, the minimum expected cost over all con- struction strategies is computed from

min j=1

�E�C�S j�� 2�

where j represents construction strategy, and m is the total num- ber of construction strategies.

The expected cost of the tunnel section �for no exploration� is

E�CNo Exploration� = min j=1

m � i=1

PiGCji �3� Decisions are made regarding the best �most cost effective� con- struction strategy�ies� based on expected values of cost given the uncertain geology.

Cost of Changing (Where Relevant) Construction Strategies between Adjacent Sections The most cost-effective construction strategy in a tunnel section, say x, may be different from the most cost-effective construction strategy in an adjacent section, say �x + 1�. Costs are therefore incurred when changing �where relevant� construction strategies between adjacent sections. These costs will be referred to as switchover costs. There are several ways in which switchover costs can be incorporated into the analyses. For example, Einstein

Table 1. Probabilistic Phase in Decision Analysis for Tunnel Exploration �after Einstein et al. 1978�

Step Probabilistic �model� phase Exploration for tunnels

1 Uncertainty in state variables Prior probability of geologic states

2 Probabilistic model Effect of geology on expected construction cost; decision tree

3 Choose among distributions Probability distributions of uncertain variables specified/assumed

4 Probabilistic sensitivity analysis Critical ranges of probabilities, and construction costs established

Table 2. Prior Probability Matrix

Geologic state

Section 1G 2G

1 0.8 0.2

2 0.4 0.6

3 0.8 0.2Fig. 2. Topography and tunnel alignment

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et al. �1987� use a dynamic programming approach to consider switchover costs when deciding on the best construction strategy at the tunnel face during construction. When using decision tree analyses, one implicitly determines the most effective construc- tion strategy in each tunnel section �since independence of sec- tions is assumed�. For example, Fig. 3 shows that S1 is the most effective construction strategy in section 2 of the tunnel shown in Fig. 2. More generally, consider part of a tunnel as shown in Fig. 4.

Let Sx * denote the most effective construction strategy in tunnel

section x, and let E�C�x,x+1�� denote the expected cost of switch- over from the most cost-effective construction strategy in section x to the most effective construction strategy in section �x + 1�. Sx

is determined by decision tree analysis, and E�C�x,x+1�� is user specified. E�C�x,x+1�� is zero if Sx

* and Sx+1 * are the same, and non-

zero otherwise. The expected switchover costs for the part of the tunnel in Fig.

4 are therefore

E�Cswitch� = E�C�x−1,x�� + E�C�x,x+1�� 4�

where E�C�x−1,x��expected switchover cost from Sx−1 * to Sx

* at the boundary between sections �x − 1� and x; E�C�x,x+1��expected switchover cost from Sx

* to Sx+1 * at the boundary between sections

x and �x + 1�. The expected switchover costs, E�C�x−1,x�� and E�C�x,x+1��, re-

flect inherent uncertainties associated with switching from one construction strategy to another, and these can be obtained, for example, from historical data and/or judgment. When determinis- tic conditions are assumed, the expected switchover costs are equal to the �specified� switchover costs.

The total expected switchover cost for the entire tunnel is ob- tained by summing up the expected switchover costs at all adja- cent section boundaries as

E�Cswitch total � = �

x=1

n−1

E�C�x,x+1�� 5�

where n�total number of sections �leading to n − 1 boundaries�.

The total expected cost of the tunnel is given by the sum of the expected construction costs in each tunnel section, and the sum of the switchover costs between sections

E�CNo Exploration total � = �

x=1

E�CNo Exploration x � + E�Cswitch

total � �6�

where E�CNo Exploration x ��expected construction cost of section x

�see Eq. �3��, E�Cswitch total ��expected total switchover costs �see Eq.

�5��, and n�total number of sections.

Information Model Phase

The information �model� phase in Fig. 1 is related to the tunnel exploration problem in Table 4. In the information phase, one repeats the tunnel cost estimation as in the probabilistic phase, but introduces the effects of �virtual� exploration into the analyses. This is explained below.

Step 1: Information Model In Step 1 of the information model, the characteristics of explo- ration �exploration strategies� are assigned. Exploration �explora- tion strategies� is characterized by its reliability defined as the probability that the exploration results indicate true conditions, and its cost. An information model, which is a decision tree, is used to determine the effects of exploration on expected costs.

Step 2: Expected Value of Information (Perfect and Sample) In Step 2, the expected value of information �perfect and sample� that results from exploration �perfect and imperfect� is determined using the results of the information model �Step 1�, and the proba- bilistic model.

Step 3: Information-Gathering Scheme In Step 3, the expected value of exploration �perfect and imper- fect� is used to determine the best information gathering scheme and devise an optimal exploration plan for the tunnel. To illustrate this, reconsider the tunnel with the alignment shown in Fig. 2, with the same uncertain geology as expressed in Table 2, and same construction costs �per unit length� shown in Table 3. Let the characteristics of exploration �exploration reliability� be as shown in Table 5.

The total expected cost of constructing the tunnel has three main components, namely, �1� the expected cost of constructing each tunnel section; �2� the expected cost of exploration; and �3�

Table 3. Construction Cost �per Unit Length� Matrix

Construction strategy

Geologic state

1G �$�

2G �$�

S1 2,100 2,150

S2 1,250 2,800

Fig. 3. Probabilistic model �decision tree� for Section 2 of tunnel in Fig. 2

Fig. 4. Illustration of most cost-effective construction strategies in different tunnel sections

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the expected cost of changing �where relevant� construction strat- egies between adjacent sections �switchover costs�.

Expected Cost of Constructing Each Tunnel Section The expected cost of constructing a tunnel section is obtained by considering each tunnel section independently. An information model �decision tree� is constructed for each section. Fig. 5, for example, shows the information model �decision tree� for section 2 of the tunnel.

There are two types of chance nodes and two types of decision nodes in the information model �see Fig. 5�. Type I chance nodes show the expected costs for a given construction strategy and given exploration results in a certain geologic state. These are computed from

E�C�S j and Exploration indicates kG� = � i=1

PiG�E�kG�Cji �7�

where iG and kG represent geologic states, S j�chosen construc- tion strategy, E�kG��geologic state indicated by exploration, PiG�E�kG� ��posterior� probability of geologic state iG given that exploration indicates geologic state kG, and Cji�cost of construc- tion strategy j in geologic state iG.

In Fig. 5, Eq. �7� results in

E�C�S1 and Exploration indicates 1G�

= �$ 1,890 � 0.6� + �$ 1,935 � 0.4� = $ 1,908

E�C�S2 and Exploration indicates 1G�

= �$ 1,125 � 0.6� + �$ 2,520 � 0.4� = $ 1,683

E�C�S1 and Exploration indicates 2G�

= �$ 1,890 � 0.1� + �$ 1,935 � 0.9� = $ 1,931

E�C�S2 and Exploration indicates 2G�

= �$ 1,125 � 0.1� + �$ 2,520 � 0.4� = $ 2,381

The posterior probability of geologic state iG given that explo- ration indicates geologic state kG is computed using Bayes’ Rule as

PiG�E�kG� = PiG PE�kG��iG

PE�kG� �8�

where PE�kG��iG�exploration reliability �from Table 5� and PE�kG��probability that exploration indicates geologic state kG, and is obtained from the total probability rule as

PE�kG� = � i=1

PE�kG��iGPiG �9�

since the geologic states are mutually exclusive and collectively exhaustive.

In Fig. 5, Eq. �9� yields

PE�1G� = �0.4 � 0.9� + �0.6 � 0.4� = 0.6

PE�2G� = �0.4 � 0.1� + �0.6 � 0.6� = 0.4

and Eq. �8� yields

P1G�E�1G� = 0.4� 0.9 0.6

� = 0.6, P2G�E�1G� = 0.6� 0.4 0.6

� = 0.4 P1G�E�2G� = 0.4� 0.1

0.4 � = 0.1,

P2G�E�2G� = 0.6� 0.6 0.4

� = 0.9

Table 4. Information Phase in Decision Analysis for Tunnel Exploration �after Einstein et al. 1978�

Step Information �model�

phase Exploration for tunnels

1 Information model Effect of exploration on expected construction cost; decision tree

2 Expected value of information �perfect and sample� Expected value of exploration �perfect and imperfect�

3 Best information gathering scheme Exploration method and configuration �geometry along tunnel�

Table 5. Exploration Reliability Matrix

Exploration indicates geologic state given reality

Reality

1G 2G

1G 0.9 0.4

2G 0.1 0.6

Fig. 5. Information model �decision tree� for Section 2 of tunnel in Fig. 2

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At Type I decision nodes, the minimum expected cost over all construction strategies for a given exploration result is computed as

min j=1

�E�C�S j and Exploration indicates kG��

= min j=1

m � i=1

PiG�E�kG�Cji �10� In Fig. 5, Eq. �10� results in values $1,683 and $1,931. The

Type II chance node shows the expected cost of imperfect explo- ration, and is given by

E�CImper f ect Exploration� = � k=1

min j=1

m PE�kG� � i=1

PiG�E�kG�Cji� �11�

Eq. �11� yields

E�CImper f ect Exploration� = �0.6 � $ 1,683� + �0.4 � $ 1,931�

= $ 1,782

The Type II chance node can also give the expected cost of perfect exploration. This is obtained using the identity matrix as the exploration reliability matrix. When this is the case, the ex- ploration reliability PE�kG��iG has a value of 1 when k and i are the same, and is zero otherwise. Eq. �11� becomes

E�CPer f ect Exploration� = � k=1

min j=1

�PE�kG�Cjk� �12�

At the Type II decision node, one can decide on whether to explore or not based on the costs of no exploration, perfect ex- ploration, and imperfect exploration. The decision needs, how- ever, to take the cost of exploration and the costs of switchover into account. These are discussed next.

Expected Cost of Exploration The cost of exploration is an evident cost that enters into the total cost of construction of the tunnel. If Cexp denotes the expected cost of exploration, then the expected cost of perfect exploration and imperfect exploration are given by

E�CPer f ect Exploration� = � k=1

min j=1

�PE�kG�Cjk� + Cexp �13�

and

E�CImper f ect Exploration� = � k=1

min j=1

m PE�kG� � i=1

PiG�E�kG�Cji� + Cexp �14�

Expected Cost of Changing (Where Relevant) Construction Strategies between Adjacent Sections The expected cost of changing from one construction strategy to another was discussed previously for the case of no exploration. Now, suppose that the information model indicates that explora- tion will be most beneficial in section x. Since, and this is impor- tant, decisions are made based on the outcome of exploration, the expected switchover costs are

E�Cswitch� = E�C�x−1,x�� + E�C�x,x+1�� 15�

where E�C�x−1,x��expected switchover cost from Sx−1 * to Sx

* at the boundary between sections �x − 1� and x, and is the weighted av- erage of the expected switchover costs with the weights being the probabilities of exploration results as

Fig. 6. Main menu of DATE Fig. 7. �a� Decision tree for no exploration in Section 2 of tunnel in Fig. 2 �see also Fig. 3�; �b� decision tree for imperfect exploration in Section 2 of tunnel in Fig. 2 �see also Fig. 5�

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E�C�x−1,x�� = � i=1

PE�iG�E�C�x−1,x�� 16�

E�C�x,x+1��expected switchover cost from Sx * to Sx+1

* at the boundary between sections x and �x + 1� and is the weighted av- erage of the expected switchover costs with the weights being the probabilities of exploration results as

E�C�x,x+1�� = � i=1

PE�iG�E�C�x,x+1�� 17�

where E�C�x−1,x��expected switchover cost from Sx−1 * to Sx

* at the boundary between sections �x−1� and x, which can be determined from historical data and/or judgment; E�C�x,x+1��expected switchover cost from Sx

* to Sx+1 * at the boundary between sections

x and �x + 1�, which can be determined from historical data and/or judgment; PE�iG��probability that exploration indicates geologic state iG, which is obtained from the total probability rule given in Eq. �9�; and n�number of geologic states.

Note that in this paper, the virtual exploration is considered in only one tunnel section at a time. As a result, the switchover costs are affected by the decision to explore �or not� in only the section �if it exists� where exploration is most beneficial. In all other tunnel sections, the switchover costs remain the same as those in the case of no exploration.

The expected total cost of constructing a tunnel section, say x, is the sum of the expected construction costs �from Eq. �13� for perfect exploration, and Eq. �14� for imperfect exploration�, the cost of exploration, and the expected switchover costs �from Eq. �15��. The cost of the entire tunnel is then given by the sum of the construction costs in each tunnel section, the cost of exploration, and the sum of the switchover costs between sections.

At the Type II decision node, the expected value of perfect information �EVPI�, defined as the difference between the ex- pected tunneling costs for no exploration and perfect exploration, and the expected value of sample information �EVSI�, defined as the difference between the expected tunneling costs for no explo- ration and imperfect exploration are computed.

Decisions are made regarding whether exploration is beneficial

Fig. 8. Expected value of perfect and sample information in all tunnel sections for tunnel in Fig. 2

Fig. 9. Simplified plan of the Sucheon Tunnel �after Min et al. 2003�

Fig. 10. Simplified cross section of the Sucheon Tunnel �after Min et al. 2003�

JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / MAY 2007 / 349

or not based on the expected value of information, perfect �EVPI�, and imperfect �EVSI�. If the expected value of perfect information is negative, then even perfect information is not cost effective. If the EVPI is positive, then one checks if the EVSI is positive. Exploration is beneficial if EVSI is positive, and the optimal ex- ploration program is the one that maximizes EVSI.

Decision Aids for Tunnel Exploration

Decision Aids for Tunnel Exploration �DATE� is a software pack- age developed in Visual Basic for Applications in Microsoft Excel that allows one to assess the consequences of collecting new in- formation through virtual exploration prior to taking any action.

The input parameters are the geologic state descriptions and their prior probabilities, the construction cost matrix, and the ex- ploration reliability matrix. The outcomes are decision trees for the cases of no, perfect, and imperfect exploration, from which the expected value of perfect information �EVPI� and expected value of sample information �EVSI� are computed. The main menu is divided into three parts as shown in Fig. 6.

In the input parameters menu, the user is able to input the prior probability matrix, the construction cost matrix, and the explora- tion reliability matrix. In the outcomes menu, the user is able to view the decision trees for each section of the tunnel. If this option is selected, submenus �see Figs. 7�a and b�� enable the user to select the section to be analyzed, and which of the no, perfect, and imperfect exploration trees to view. Figs. 7�a and b� are the decision trees for no exploration and imperfect exploration in Section 2 of the tunnel that was analyzed previously �and shown in Fig. 2�. The scenario manager allows the user to load previ- ously saved scenarios, and save current ones. Further details on DATE are provided in Karam �2005�.

The “Select Section” submenu in the top right corner of Figs. 7�a and b� allows the user to choose the desired section to be analyzed. By doing so, similar decision trees as those in Figs. 7�a and b� can be constructed for the other tunnel sections.

The output can be displayed graphically as shown in Fig. 8, which shows the expected value of perfect and sample informa- tion in each tunnel section.

Note that the cost of exploration is not included in the results in Fig. 8 for illustrative purposes. Fig. 8 shows that the expected value of sample information is greatest in section 2 of the tunnel. Assuming that the value is greater than the cost of exploration, the decision would be made to explore in this section.

Table 6. Description of Geologic States

Geologic state

Description

RMR Resistivity ��m� Q Value

1G �81 �3000 �40

2G 80–60 1000–3000 4–40

3G 40–60 300–1000 1–4

4G 20–40 100–300 0.1–1

5G �20 �100 �0.1

Table 7. Description of Construction Strategies

Table 8. Tunnel Sections and Prior Probability Matrix

Section

Geologic state

Station Length �m� 1G 2G 3G 4G 5G

1 9 + 340 65 0.51 0.49 0.00 0.00 0.00

2 9 + 405 20 0.15 0.47 0.38 0.00 0.00

3 9 + 425 57 0.15 0.47 0.38 0.00 0.00

4 9 + 482 40 0.00 0.00 0.47 0.53 0.00

5 9 + 522 120 0.00 0.29 0.64 0.07 0.00

6 9 + 642 41 0.15 0.25 0.53 0.07 0.00

7 9 + 683 96 0.14 0.25 0.56 0.06 0.00

8 9 + 779 104 0.47 0.43 0.10 0.00 0.00

9 9 + 883 586 0.48 0.45 0.07 0.00 0.00

10 10 + 469 106 0.49 0.51 0.00 0.00 0.00

11 10 + 575 67 0.51 0.49 0.00 0.00 0.00

12 10 + 642 167 0.50 0.50 0.00 0.00 0.00

13 10 + 809 36 0.78 0.22 0.00 0.00 0.00

14 10 + 845 74 0.10 0.83 0.07 0.00 0.00

15 10 + 919 43 0.19 0.81 0.00 0.00 0.00

16 10 + 962 110 0.67 0.33 0.00 0.00 0.00

17 11 + 072 188 0.86 0.14 0.00 0.00 0.00

18 11 + 260 50 0.13 0.32 0.54 0.00 0.00

Table 9. Construction Cost �per Unit Length� Matrix

Construction strategy

Geologic state

1G �$�

2G �$�

3G �$�

4G �$�

5G �$�

S1 3,150 4,500 5,700 7,200 8,850

S2 4,125 3,525 5,325 7,350 9,000

S3 4,350 4,875 4,650 6,375 8,138

S4 4,650 4,500 5,400 6,285 7,875

S5 4,425 4,575 5,175 6,450 6,600

Table 10. Exploration Reliability Matrix

Exploration indicates geologic state given reality

Reality

1G 2G 3G 4G 5G

1G 0.9 0.1 0 0 0

2G 0.1 0.9 0.1 0 0

3G 0 0 0.8 0.1 0

4G 0 0 0.1 0.8 0.1

5G 0 0 0 0.1 0.9

Case Study: Sucheon Tunnel In this section, DATE is used to evaluate the optimal exploration plan for the Sucheon Tunnel in South Korea �Min et al. 2003�. The Sucheon Tunnel is a 2-km tunnel located mainly in micro- graphic granite and diorite. Fig. 9 shows a plan view of the tunnel alignment and Fig. 10 shows a simplified cross section.

The geology of the tunnel was investigated using different methods including borehole drilling, electrical resistivity surveys, and seismic exploration. From this information, five geologic states were identified based on RMR, resistivity, and Q value, as

shown in Table 6. Five construction strategies were available to the contractor. These are labeled S1 to S5 and are described in Table 7. The tunnel was divided into 18 sections, as shown in Table 8. Using the geologic information and engineering judg- ment, the prior probabilities of each geologic state in each tunnel section were assigned. These are shown in the prior probability matrix in Table 8. The construction cost matrix �costs per unit length� is shown in Table 9, and the exploration reliability matrix is shown in Table 10. The DATE package was then used to assess the value of exploration in each tunnel section. Decision trees for

Fig. 11. �a� Decision tree for no exploration in Tunnel Section 9; �b� part of decision tree for perfect exploration in Tunnel Section 9; and �c� part of decision tree for imperfect exploration in Tunnel Section 9

no, perfect, and imperfect exploration were constructed for each tunnel section. Fig. 11�a� shows the decision tree for no explora- tion in section 9 of the tunnel. Figs. 11�b and c� show parts of the decision trees for perfect and imperfect exploration in section 9 of the tunnel, respectively. In particular, they show the branches of the tree when exploration indicates geologic state 2G. The entire trees include similar branches for the cases when exploration in- dicates the other geologic states, i.e., 1G, 3G, 4G, and 5G. Note that when constructing the perfect exploration tree, the identity matrix is used as the exploration reliability matrix in Table 10. As a result, the probabilities of encountering the geologic state indi- cated by the exploration are equal to 1, whereas the probabilities of encountering all other geologic states are zero �see Fig. 11�b��. When constructing the imperfect exploration tree, the exploration reliability matrix in Table 10 is used, and the probabilities of encountering other geologic states than what exploration indicates are nonzero �see Fig. 11�c��. Fig.12 shows the expected values of perfect �EVPI� and sample information �EVSI� in all tunnel sec- tions. The effects of exploring in different tunnel sections can also be shown as in Fig. 13, which shows the expected cost of tunnel- ing in each section for the cases of no exploration and �imperfect� exploration.

Figs. 12 and 13 show that exploration is beneficial in tunnel sections where the expected value of sample information is greater than the cost of exploration. Assuming a cost of explora- tion of $30,000, Fig. 12 indicates that exploration is beneficial in Sections 5, 7, 8, 9, 10, and 12. For a constant cost of exploration, exploration is most beneficial in section 9, where the expected value of sample information is greatest. Note that this is also the longest section of the tunnel, and where the overburden is greatest �see Fig. 10�. The decision is therefore made to explore in section 9. This is the optimal exploration location for one boring. Explo- ration will result in expected savings in the tunnel cost equal to the expected value of sample information minus the cost of ex- ploration, i.e.

E�Savings� = EVSI − Cexp = $ 245,500 − $ 30,000 = $ 215,500

Based on the results of the exploration, the prior probability matrix can be updated, and the analyses redone to determine whether further exploration is beneficial, and what the optimal location is. Successive repetition of this process will lead to the development of an optimal exploration plan for the tunnel.

It is worth noting that the analyses presented in this paper have been simply based on cost. A more general analysis could be done in terms of utility analyses, which would allow one to consider the decision maker’s attitude toward risk �e.g., see Keeney and Raiffa 1976�. It is important to note that the procedures described in this paper are also applicable when utility analyses are used by replacing costs with utilities.

Conclusions

Exploration planning is a process of decision making under uncertainty. This paper, after briefly reviewing the decision ana- lytical procedure for tunnel exploration, provided practical tech- niques to do so. A software package developed in Visual Basic for Applications in Microsoft Excel the Decision Aids for Tunnel Exploration �DATE� was developed for this purpose. DATE was then used to devise an optimal exploration plan for the Sucheon Tunnel in Korea. Decision aids, such as DATE, allow one to asses the value of collecting new information through virtual explora- tion prior to taking any action.

Acknowledgments

The writers would like to acknowledge that the data for the Su- cheon Tunnel were made available by the Korean Railroad Insti- tute �KRRI� and SK Engineering & Construction. In particular, the help of Dr. J.-S. Lee in this regard was instrumental. The authors would also like to thank the anonymous ASCE reviewers for their many useful comments and suggestions which led to a significant improvement of the paper.

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