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Decision analysis applied to rock tunnel exploration 1978 baecher

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  • 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).
  • 4. 146 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
  • 7. 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.
  • 9. 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
  • 11. 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.
  • 12. 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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