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Optimization of underground water transmission network systems under seismic risk
Optimization of underground water transmission network systems under seismic risk R. T A N and M. S H I N O Z U K A Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, N. Y 10027
The purpose of this paper is to establish procedures for the optimum design of underground water transmission systems to be newly constructed as well as for the optimum improvement of existing systems on the basis of seismic risk and cost-effectiveness considerations. As an example, a realistic water transmission system will be analyzed and optimized. Procedures for optimizing the total cost are developed in which the pipe diameters and the network topology are parameters which can be varied to achieve a minimum total cost under certain realistic constraints. The total cost includes the initial cost premium, repair costs and incident (fire) costs. The optimization algorithm employed in this study is derived from the discrete gradient method.
INTRODUCTION In a recent paper 1, Shinozuka et al developed a methodology for the assessment of the seismic risk which underground water transmission systems in seismically active areas are subjected to, in terms of the degree of possible unserviceability resulting from destructive earthquakes. In that study, it was postulated that a water transmission system was serviceable when its firefighting capabilities (measured in terms of water pressure and flow rate at specified nodes) remained intact immediately following an earthquake. With the aid of techniques developed by Shinozuka et al in previous papers 2 - 5, the probabilities of link failure were evaluated and Monte Carlo techniques were used to generate a sample of simulated states of damage for the water transmission network. The Newton-Raphson method was then utilized to perform a flow analysis on each of these damaged networks and, at the same time, to determine whether or not it still remained serviceable. A numerical example was presented to demonstrate the probabilities of system unserviceability estimated under a set of assumed parameter values deemed reasonable. However, the construction or improvement of water transmission systems usually involves a large capital outlay and continuous expenditures for maintenance and repairs that are only balanced by the various benefits the systems can generate. Therefore, it is necessary to perform a "seismic design decision analysis" on such systems in order to examine the advisability of alternative design decisions on the basis of the cost-effectiveness of the feasible strategies for the design, construction and improvement of those systems subjected to seismic risk. The purpose of this paper is to develop a design and improvement optimization methodology within the framework of such a seismic design decision analysis. Achieving this purpose is considerably difficult due to the complex operational as well as structural functions the network system must perform to satisfy the prescribed serviceability criteria, e.g., water flow rate and pressure
requirements. The difficulty is compounded even further because of the wide range of choices available for such design and improvement strategies involving such inputs as the pipe size, pipe material, joint structure, buried depth, material and construction method of back fill, pipe routing, seismic magnitude to be used in the design, etc., under competing constraints on the system's reliability and economics. Some recent papers by Lam 6, Watanetada 7 and Cenedese and Mele a have considered such optimization problems, specifically those dealing with water transmission and distribution systems without, however, taking into account the potential seismic destruction of the system. At the same time, more general aspects of the optimum design decision analysis for engineering systems in seismic environments with strong socioeconomic implications have also attracted increasingly serious attention among engineers; for example, Whitman et al 9, Blume and Monroe 1°, Grandori and Benedetti ~1, Liu and Neghabat 12 and Shah and Vagliente ~3. Not much work in this direction with respect to underground water transmission network systems has been pursued, however. In the present study, the optimum design of a water transmission system is derived on the basis of a design decision analysis involving cost-benefit considerations so as to balance the initial cost of the design, construction and improvement of the system and the future losses that could accrue from earthquake-induced malfunctions of the same system. Such losses are usually classified into two categories: the direct loss, somewhat easier to evaluate, results from such post-earthquake activities (necessitated by the unserviceability) as emergency and permanent repairs of the seismic damage sustained by the system, emergency water delivery, etc. On the other hand, the indirect loss consists of inconveniences suffered by residential users, damage due to postearthquake fires, industrial loss, etc., pertaining to the malfunction or unserviceability of the system. The indirect loss is therefore much more difficult to assess.
0261 7277/82/01(X)3(~ 0952.00 © 1982 CML Publications
30
Soil Dynamics and Earthquake Engineering, 1982, Vol. 1, No. 1
Optimization of underground water transmission network systems: R. Tan and M. Shinozuka The specific optimization methodology to be developed in the sequel is based on an analysis that examines the sum total of the initial cost and the (indirect as well as direct) future losses as a function of the design or improvement strategy and recommends a strategy which will achieve the optimum cost. The step-by-step procedure of such an analysis can be delineated as follows: (1) a mathematical model of the water transmission network is constructed. (2) A cost or objective function is introduced as a function of the pipe length and diameter and the expected seismic losses. (3) The discrete gradient method is employed to find the solution that minimizes the cost function under appropriate constraining conditions involving the system performance criteria.
NETWORK TOPOLOGY In dealing with a water transmission system to be newly constructed, the present study assumes that the locations of the source nodes (water supply stations) and demand nodes are predetermined. Therefore, finding their optimum locations is not part of the analysis to be performed here. However, it is acknowledged that the siting problem associated with these source and demand nodes, while not specifically considered here, is essentially that of overall system optimization, and its solution requires careful consideration not only of the spatial as well as temporal distributions of the water demands for domestic, industrial and other consumption, but also of such environmental constraints as the topographical and geological conditions of the general area in which the transmission system is to be constructed. With respect to the links (or pipelines) that will connect these source and demand nodes, their routing also requires intelligent techno-economic judgment in order to select only those options that are viable and feasible from the viewpoint of the construction costs. Those routes that would be in direct conflict with other major structures, existing or planned, and that require passage through private property and/or regions under unfavourable topographical, geological and soil conditions, should be considered unfeasible. Those routes that would produce a redundancy in the transmission network only at an obviously disproportionate cost should also be considered unfeasible. Thus, the decision on the routing of links cannot be made as arbitrarily as might appear. For this reason, the present study performs an optimization analysis on a network for which not only the nodes but also all possible routes for the links are prescribed. In fact, Figure l(a) illustrates such a network in which a dashed line connecting two nodes implies a feasible pipeline construction between them. If there is no dashed line indicated between two nodes (e.g., between nodes 3 and 5), it means that the construction of a pipeline between these two nodes has been ruled out on the basis of infeasibility (but not of optimality). As mentioned above, the infeasibility may be due to (a) interference with other major structures, (b) unfavourable topographical, geological and soil conditions and (c) obvious uneconomical redundancy. Under these circumstances, the diameter of the pipes (assumed to be uniform at least within each link) becomes the only parameter that can be varied to achieve the optimum design of the system. Indeed, the optimization method to be described later will indicate which possible
routes should or should not be actually constructed for the optimum design. It will also indicate the optimum diameter of the pipe in each of the links to be constructed. The total number of links, therefore, may be decreased but never increased as a result of such an optimization analysis. For example, the analysis may find that an optimum design is achieved if all the feasible links in Figure l(a) except for links 1 3, 6-8 and 5-8 are constructed, thus resulting in an optimum network as shown in Figure l(b). It is not uncommon to find that an optimally designed water transmission system, e.g., Figure l(b), becomes inadequate after a period of time due to the growth of population and the increased industrial activities within the community served by the system. In order to meet these new demands, the system must be expanded by either replacing the old pipelines with those possessing a larger capacity or adding new pipelines to the network system along feasible routes that were not used for the optimum design: routes 5-8, 6-8 and 1-3 in Figure l(a). With respect to the replacement of old pipelines, however, it is often more economical to construct a new pipeline parallel to an existing one. This practice is known as "looping" and it obviously results in increased link capacity. Figure 2 shows a possible expansion of the network in Figure l(b) in response to increased demands at nodes 3 and 5. The expansion reconsiders not only the possible use of those feasible routes that were not part of the initial optimum design (routes 5 8, 6 8 and 1-3) but also the looping of all existing links. Hence, in Figure 2, solid lines indicate existing pipelines, while dashed lines possible new pipelines. The optimization analysis will then determine which new pipelines should or sl~ould not be constructed.
9
9 I
I
',
I
-
~5
,~ I
\
I
I
I
I
"-6 .....
-6,4
\
7(S.
~l
i
8
8
(a)
I0
10
(b) source node
0
demand node link feasible link
Figure 1. Newly constructedsystem.
Soil Dynamics and Earthquake Engineering, 1982, Vol. 1, No. 1 31
Optimization of underground water transmission network systems: R. Tan and M. Shinozuka
9
92 3 7
•
% node with new
I0
demand
A
source node
0
demand node existing link
---
Fioure 2.
feasible new l i n k
Expanded system.
COST F U N C T I O N S The initial cost function CFx, which includes the cost of the pipes themselves and the cost of construction, represents the initial cost premium. The cost CFou per unit length of the pipeline between nodes i a n d j depends on the pipe diameter D~jk (uniform throughout its length Lij) and other factors unique to link ij (e.g., soil conditions along the route). The subscript k herein refers to the pipe size number. Using a generally accepted monomial formula, the cost CFo~j is then approximated by CFo~j = w~jD~jk, where 7 is a nondimensional constant between 1.0 and 2.0 and w~j is the so-called cost index for the pipeline ij which reflects the constrqction as well as pipe costs. Then, the initial cost premium can be expressed by
NONO CFx = Y, ~', wijD~jkLijtij i j
i , j = l , 2, .. ND i>j "'
(1)
where L u = the length of link ij connecting nodes i and j, ND = the total number of demand and source nodes and tij = the topological index: tij = 0 if link ij is unfeasible and tij = 1 if link ij is feasible. For example, in Figure l(a), t35 =0, t24 =0, t16 = I, t58-- l, etc. The present optimization analysis permits the pipe diameter to vary within a set of discrete values: Dij o =0, Dijt = 101.6 cm (40'3, Dij2 = 114.3 c m (45"),..., Dij 9 = 203.3 cm (80"). If the analysis indicates that t~j = 1 but D~jk= 0 (or k = 0) for an optimum design, it means that route ij is feasible but the optimality requires that it not be constructed. The thickness of the pipe is assumed to be 16 mm, regardless of the diameter size. As mentioned earlier, in addition to the initial cost premium CF1, the direct and indirect losses resulting from earthquakes must enter the cost-benefit formulation. With respect to direct loss, only the repair cost will be considered in the present study for the sake of simplicity.
32
As to the indirect cost, only the loss due to postearthquake fire will be considered in order to be consistent with the serviceability criteria postulated and investigated 1. The disasters associated with the 1906 San Francisco earthquake and the 1923 Tokyo earthquake warrant such a study dealing with the post-earthquake fire-fighting capability of water transmission systems. Indeed, in the case of the San Francisco earthquake, the damage directly attributable to ground shaking was estimated at $24 million while the loss due to the fire caused by the earthquake was $500 million (in terms of 1906 dollars). Also, the Tokyo earthquake claimed 143,000 human lives primarily due to the fire that immediately followed the earthquake. The optimization procedure to be used here is based on the expected total cost to be estimated with the aid of a damage probability matrix for each link and an unserviceability probability matrix for each node. Table 1 shows the damage probability matrix for link ij and its nm element indicates P[Lijk,lm], the probabilities of link ij in the damage state n, given that the earthquake magnitude M = m. Note that the event L~jkl = the state of link ij in major damage, or equivalently, failure of link ij, L~k3 = the state of link ij in minor damage, or survival of link ij and L~jk2 = the state of link ij in moderate damage. These probabilities were obtained from reference 1. Similarly, the unserviceability probability matrix at node i is shown in Table 2. Its n m element indicates P~(n[m) where Pi(llm), Pi(2lm) and Pi(3lm) are written for Pi(majorlm), Pi(moderatelm) and Pi(minorlm), respectively. Each element refers to the probability of minor, moderate or major unserviceability associated with node i, given that the earthquake magnitude M = m. As mentioned ~, a state of major unserviceability corresponds to the condition that at least either the water pressure or the flow rate is less than the respective minimum requirements (Emi n for pressure and Qmin for flow rate) for firefighting. A state of moderate unserviceability represents the condition under which both the pressure and flow rate satisfy the minimum requirements but do not simultaneously exceed the design allowable amounts associated with the undamaged original network (Er.od for pressure and Qmod for flow rate). A state of minor
Table 1. Damage probability matrixfor link ij Damage states, n 1. Major 2. Moderate 3. Minor
Central damage ratio
CDR1 CDR2
Earthquake magnitude, m 5
6
7
8
9
P[LisR. Im]
CDR3
Table 2. Unserviceabilityprobability matrix for node i Earthquake magnitude, m Unserviceability U n s e r v i c e a b i l i t y 5 6 7 8 9 states, n cost ratio 1. Major 2. Moderate 3. Minor
Soil Dynamics and Earthquake Engineering, 1982, Vol. 1, No. 1
UCR~ UCR2 UCR 3
Pi(nqm)
Optimization of underground water transmission network systems: R. Tan and M. Shinozuka unserviceability is then defined as that in which the pressure and flow rate exceed Emod and Qmod, respectively, and therefore implies a state of normal serviceability. These probabilities were also obtained from reference 1. The second column of Table 1 lists the central damage ratio CDR, as a representative value of the ratio of the repair cost of the pipeline (between nodes i and j) in the damage state n to its replacement costs 9'14. Usually, primarily due to the paucity of relevant information, reasonable values are selected for these ratios essentially on the basis of engineering judgment. Utilizing the central damage ratios thus introduced, the expected value of the repair cost CF2 for a period of T years can be written in approximation as
ND ND 3 ECF2 = ~ Z wijDi]kLi/ij x T Z CDR. x P[Lijk,] x DC i
j
n=l
i>j
(2)
capability of the water transmission system would, on the average, be greater than those under that of an intact system. As far as the indirect losses are concerned, it is this incremental component of the fire losses that this study addresses itself to. Therefore, the second column in Table 2 lists the unserviceability cost ratio, UCR., which reflects such an incremental component of fire losses and is defined below: Consider a representative value of the ratio of the fire losses that would be incurred if the water transmission system is in the damage state n immediately following an earthquake to the fire losses if the system remains intact. Subtract unity from this ratio. This then is the unserviceability cost ratio pertaining to the fire losses. This ratio is perhaps even more difficult to estimate than the central damage ratio. For the purpose of numerical analysis, however, it is assumed that U C R 3 = 0, U C R 2 = 2 and UCRI=4. The expected value of the indirect losses C F 3 due to fire for a period of T years can then be written as
with
N' 3 E C F 3 = T E E EFCi × UCR, x P~(n)× DC
/5[L,~k,] = v ~ P[ giik,lm] {F M(m+ 0.5) -- F M(m-- 0.5)} (3)
(5)
i=ln=l
m
where w~jD~jkLgjhas been interpreted as the replacement cost, v is the annual rate of earthquake occurrence, DC is a discount factor, and FM(') is the probability distribution function of M: FM(m) = { 1 -- [ e x p -- fl'(m -- mL) ] }/{ 1 - e x p [ - fl'(m u - m L)] } (4)
where m L is the lower bound of the magnitude value of engineering significance ( = 4.0 in the present study), while mu is the largest magnitude value expected from the earthquake, and fl' is a site-dependent constant. In this study, mu=9. Thus, the product v{FM(m+O.5)-FM(m -0.5)} indicates the annual occurrence probability of an earthquake with a magnitude between m - 0 . 5 and m + 0.5. The quantity P[Li~k,] in equation (2) represents the expected value of the annual failure probabilities of the pipeline between nodes i and j in the damage state n and
with
P,(n) = v ~ P,(n[m){FM(m+ 0.5) -- FM(m-- 0.5)}
(6)
m
where N' is the total number of demand nodes. Also, in equation (5), EFCi represents the expected fire incident costs for the area that is covered by node i as protection against fire when the water transmission system survives an earthquake without damage. As such, EFCi depends on the characteristics of each region represented by the type of buildings and structures, residential, commercial or industrial, to be found therein. Again, it is rather difficult to estimate such a quantity without recourse to subjective judgment. Finally, the expected total cost E T C F can be constructed as
ECTF = CF1 + ECF2 + ECF3
3
CDR, x P[Lijkn ] is the corresponding expected value n=l
of the annual damage ratio. There is always the possibility of a post-earthquake fire and once such a fire occurs, it will inflict a certain amount of loss upon the community in general. The amounts of such losses (categorized as indirect losses in the present study) depend on a number of inherently probabilistic factors: for example, (a) exactly when, where and how it occurs and under what weather conditions, (b) to what extent the community fire-fighting force remains effective in terms of the availability of manpower and fire engines and in view of the accessibility of the fire engines to those areas of conflagration. Such availability and accessibility can be seriously hampered by the possible destruction of the fire engine stations, bridges, highways, etc. and by the blockage of streets by collapsed structures and debris. And of more relevance to the present study, (c) how serviceable the water transmission system remains immediately after the earthquake in terms of its firefighting capability. Particularly with respect to item (c) above, the losses due to fire under the degraded conditions of fire-fightlng
NONO / + = x DC) = ~i ~j wijD~'jkLijtlj~ 1 T ~ , CDR,× "[Lijkn]
N' 3 + T ~ ~, EfC, x UCR. x Pi(n) x DC i=1
i>j
(7)
n=l
where P[Li~R,] and Pi(n) are given by equations (3) and (6), respectively. The preceding formulation of the expected costs, particularly in terms of equations (2), (5) and (7), applies to water transmission systems to be newly constructed. However, the same equations can be employed for expanding a system, with the following modifications: (a) Only the initial cost premium of the new pipelines is considered for CF1 and (b) ECF2 includes only those repair costs for the new pipelines. Several economic criteria such as the net present worth, annual equivalent value and benefit-cost ratios 15 can be used by designers and decision makers to rank alternative design and improvement strategies and to select the best among them, provided that all the costs and benefits associated with each strategy can be estimated. In the
Soil Dynamics and Earthquake Engineering, 1982, Vol. 1, No. 1 33
Optimization of underground water transmission network systems: R. Tan and M. Shinozuka 2B
~2
¢xl
2 "7.
I
-----~direction
of steepest
descent at point I
• /
~2A
2Cj
g "7,
it_
Decision Variable I
Figure 3. Steepest descent in discrete decision variable space. present study, however, the expected total cost function E T C F in equation (7) is considered the objective function to be minimized for optimization with respect to the decision variables D~jk under the following constraints at all the nodes:
Ei(Mo) >1Emin
i = 1, 2..... N'
(8)
Oir(MD) >~Qmin
i = 1, 2..... N'
(9)
where E,i(Mo) and Qi,(Mo) are, respectively, the conditional expectations of El, the water head at node i, and Q~,, the rate at which the water is extracted from node i, given that the earthquake magnitude is equal to the design magnitude Mt~ These conditional expectations can be obtained from reference 1. In the present study, selecting a specific value for the design earthquake magnitude M o from among the various possible alternatives (say, M o = 5, 6, 7, 7.5 and 8) is considered, for simplicity, to represent the entire design strategy, although in reality the design strategy implies a set of feasible choices based on a number of pertinent design parameters. Therefore, the constraining conditions given by equations (8) and (9) are of particular importance in view of the fact that it is only through these conditions that the design earthquake magnitude is reflected in the design or in the choice of optimum pipe diameters. DISCRETE GRADIENT METHOD The gradient method is used to minimize the objective function E T C F in equation (7) under the constraints of equations (8) and (9). As is well known, this method finds the path of steepest descent along which the numerical iteration can proceed to eventually locate a local minimum 16. Since the present decision variables D~jk and hence the objective function are discrete, it is not possible to evaluate in the usual sense the gradient of the objective function from which the direction of steepest descent can be found. However, this difficulty is not a serious one and can be circumvented by the use of the interpolated objective function which is defined over a continuous range of pipe diameter values. Further difficulties lie in the fact that (a) the objective function E T C F in equation (7) involves the probabilities of link failure and unserviceability in various states of damage which are extremely complicated functions of the decision variables D~jk and (b) the direction of steepest descent will not necessarily pass through a feasible point in the decision variable space due to the discreteness of the decision variables. Figure 3 schematically illustrates this
34
for the case involving two decision variables. To overcome difficulty (a) above, the direction of steepest descent for the initial cost premium CF1 given by equation (1) will be used in approximation as the direction of steepest descent associated with the expected total cost function E T C F in the iterative procedure of finding the local minimum. Such an approximation will drastically reduce the amount of numerical effort to be otherwise expended in such an optimization procedure since, in this way, the direction of steepest descent can be found analytically. Obviously, the approximation will produce a reasonable result if the behaviour of the sum of the expected repair costs ECF2 and the expected incident (fire) costs ECFa, with respect to variation in the decision variables, is similar to that of the initial cost premium. This is numerically confirmed to be the case in examples considered later. Difficulty (b) above can be avoided by following the suggestion made by Lam 6 on the basis of empirical results. The cosine of the angle 0ij between the direction of steepest descent and the axis of the decision variables Dijk can be shown to be cos Oil= -
/~
,~,ij,lijk ,~ij,ij
i >j
(10)
4~i ~ ('wijD[~klLutu)2 Lam's procedure utilizes these cosine values in the following fashion: Suppose, at a particular cycle of iteration, that a certain number of pipelines have cosine values larger than 0.5; cos0ij~>0.5. In this case, (a) the diameters of these pipelines will be decreased from Dijk to Dij~k- 1,- These reduced values of pipe diameters, together with the ones unchanged, represent a new decision point along the path of steepest descent. (b) A check will be made to see if this new decision point is a feasible point in terms of satisfying the constraining conditions. (c) If it is a feasible point, the path of steepest descent will be extended further from this point and the process of optimization will continue. (d) If it is not, the diameter of the pipeline with the largest cosine value will be reduced from D~jk to Dij(k 1, thus establishing a new decision point. (e) A check is made to examine if it is in the feasible domain. (f) If it is, the path of steepest descent is extended and the iteration proceeds. (g) If it is not, the diameter of the pipeline with the second largest cosine value will be reduced, by one size, the feasibility will be checked, and the procedure will be repeated until a new feasible decision point is found. It may happen, however, that no new decision point can be found in the feasible domain by adjusting the diameter of the pipelines with cos 01j >~0.5. If this happens, then (h) the reduction (by one size) of the diameter of those pipelines with cos 0~j < 0.5 will be considered one by one in such an order that the pipeline with the larger cosine value will be considered before one with a smaller cosine value. (i) If none of the pipelines involved produces, upon reduction (by one size) of its diameter, a new decision point in the feasible domain, the current decision point represents the local (and possible global) optimum design. (j) The above procedure results, in general, in a different local minimum, depending on the initial decision point of the initial values that must be assigned to the decision variables for the purpose of starting the iteration. If such initial values produce cos0~j which are smaller than 0.5 for all the pipelines involved, then the procedure should begin at step (h).
Soil Dynamics and Earthquake Engineering, 1982, Vol. 1, No. 1
Optimization of underground water transmission network systems: R. Tan and M. Shinozuka Table 3. Scheme for findin9 the 91obal minimum initial design
links "--....point with
~
larger sizes
1"
8
9
0
=
,-
E ~
2
3
4
0
0
5
6
7
smaller sizes
0 +
+
0 + +
intermediate sizes
+
+
+
* indicates iteration + indicates -- indicates 0 indicates
0
0
+
10
8
the use of the largest available size for each link starting the increasing the pipe size by one level; Dqk~Dq(k+ ~j decreasing the pipe size by one level; Dok~Di~(k 17 no change
While the procedure described above certainly yields a local minimum, there is no practical method that can be used to locate the decision point corresponding to the global minimum for the present optimization problem. Therefore, the present study uses a standard technique in which the global minimum is estimated as the minimum of a number of local minima resulting from iterations initiated at different sets of initial decision points. In the present study, these initial decision points are not specified at random in the decision variable space as is often done when solving this type of optimization problem, but by means of a systematic scheme as illustrated in Table 3. This scheme economically generates the initial decision points, thereby increasing the possibility that the minimum of the resulting local minima will indeed be the global minimum. The second column of Table 3 corresponds to an optimization process starting at the first initial decision point which represents the use of the maximum available pipe size for all the pipelines. This first initial decision point will result in the first minimum value of the objective function and the corresponding first optimum design. The third column of Table 3 then indicates an optimization process starting at the second initial decision point which is obtained by adjusting the first optimum design. This adjustment consists of increasing, by one size, the diameter of the intermediately sized pipelines (intermediate in diameter size) in the first optimum design. This second initial decision point will lead to the second minimum value of the objective function and the corresponding second optimum design, each of which may or may not be the same as the corresponding quantities associated with the first decision point. If the second minimum value of the objective function is smaller than the first minimum, the third initial decision point is obtained from the second optimum design by increasing (by one size) the diameter of the smaller sized pipelines in the latter. If not, the third initial decision point is derived from the first optimum design by increasing (by one size) the diameter of the smaller sized pipelines in the first optimum design. This procedure continues in accordance with Table 3 which indicates how the new initial design points can be systematically generated by adjusting the optimum design points already obtained. The numerical examples that follow will show that the number of different initial decision points needed to track down the global minimum is usually less than ten. N U M E R I C A L EXAMPLES Two examples are presented: In the first example, the optimum design of a water transmission system is
considered, while in the second, the optimum expansion of the transmission system optimized in the first example is demonstrated. In fact, the water transmission network to be considered herein is a simplified version of the Los Angeles system and consits of three (3) supply stations, ten (10) demand nodes and eighteen (18) links, as shown in Figure 4. The physical data to be used in these examples are the same as those used in reference 1. Those commercially available sizes for pipe diameters are assumed to be those corresponding to k = 1, 2. . . . . 9 in the first example. The diameters of the pipelines to be added to the existing system in the second example are those corresponding to k = 0, 1, 2 ..... 9. This implies that in the first example, all the links shown in Figure 4 must be constructed and thus the optimum design maintains the same topological characteristics as those in Figure 4 since k---0 is not included an an option in this example.
Optimization of a system to be newly constructed As mentioned above, the water transmission system depicted by in Figure 4 is considered. By assumption, all the links shown are feasible and must be constructed (k :~ 0). Also, it is assumed that 7 = 2.0, wo = 1 and the five design strategies, l-V, respectively, represent the design earthquake magnitudes MD= 5, 6, 7, 7.5 and 8. Figure 5 indicates the variation of the optimum initial cost premiums, under strategies I (curve A) and III (curve B), computed for different starting decision points under the scheme illustrated by Table 3. The minimum values among these local optima are observed for starting decision points 6, 8, 9, 11 and 12 under Strategy I. These values are identical within a small range of computational error and are considered as the global minima under this strategy. Similarly, under Strategy lII, the global minimum is estimated as the local optima associated with starting decision points 9 and 12. For ease of comparison, these values are plotted in Fig. 5 in terms of a percentage of the global minimum value under Strategy I. Also in Figure 5, the numbers in parentheses indicate those of the iterations needed to reach the corresponding optimum values. Both curves A and B suggest that the local minimum, which will eventually be taken as the global
13
1 supply station demand node 2
3
12[
11
Figure 4.
Water distribution system of Los Angeles.
Soil Dynamics and Earthquake Engineering, 1982, Vol. 1, No. 1 35
Optimization of underground water transmission network systems: R. Tan and M. Shinozuka (24)
(]75) (i81
(27) -~C~ve
B for Strategy III (19)
(23)
114 112(83)
g g T~
\-
Ii0~(23
)
I 0 8 ~ 106
[numbers in parentheses indicate the number of iterations for convergence]
required
(26) (23)
o/L
~
(23)
104
O) Curve A for
2
102 Strategy I
(2)
(24)
conservative strategy is adopted, the repair costs increase but their relative percentage values decrease since the initial cost premiums increase more rapidly. As shown in Figure 7, however, the incident fire costs respond to conservative strategies much more favourably by exhibiting a sharper decrease per larger initial cost investment. The expected total costs are indicated by curve B in Figure 6 which concludes that design strategies I and II share optimality. The corresponding diameter sizes of the pipelines are shown in Figure 8 in terms of k values.
(15) 2'
3
4
5
6
~
8
9
1'0
1'I
1'2
13
starting points for iteration
Figure 5. Initialcostpremiumford~erentstartingpoints.
Optimum expansion of an existing system The system optimized in the preceding example and shown in Figure 8 is considered the existing system upon which an increased demand has been imposed. The new
o initial cost premium(CF1) expected total cost 300
(ETCF) .16
°r(,'-,,i
25O
oexpected repair cost n expected incident(ECF2)/~ cost (ECF3/ 1
2OO
+,
-14 C u r v e B ~ ~ /
150
CuI
.12
> ,m'-
m-'--
Curve A
100 I
I,II III strategy
I
IV
0"I
I
V
I
[(I+i) T - 1]/[Ti(1 +i)~
I
I,II III strategy
Initial cost premium and expected total costs.
DC=
(8.2)
(8.7)
8
minimum appears before the tenth initial decision point is tried. The global minima of CF1 associated with the five design strategies I V are plotted in Figure 6 (curve A) while the corresponding expected repair costs ECF2 and the expected incident (fire) costs ECFa are shown in Figure 7 for the same design strategies. The expected repair and incident costs are computed under the assumptions that (a) CDR 1,2,3 =0.9, 0.4, 0.0, (b) UCRI.2.3 = 4, 2, 0, (c) the design life of the system = fifty (50) years, (d) the annual rate of earthquake occurrence v = 3, (e) the expected fire incident costs EFCI for the area that is covered by demand node i as protection against fire in an undamaged system are 1.33~o, 5.32~o, 5.32~o, 1.33°~o,2.66~ , 3.990, 6.65~o, 3.99~o, 6.65~0, 2.66% of the initial cost premium CFx of the system, respectively, for i = 1 - 10, (f) the discount factor is given as
I
I
IV
V
Figure 7. Expected repair and incident costs. k=8
I-l supply node 6
1
J
(11)
with the discount rate i=8~o. The numbers in parentheses in Figure 7 indicate percentage figures of the repair costs with respect to the initial cost premiums under the same design strategies. When a more
36
/~ \
-10 I
Figure 6.
(6"4)
Figure 8.
Soil Dynamics and Earthquake Engineering, 1982, Vol. 1, No. 1
Optimum pipe sizes.
0
demand node
Optimization of underground water transmission network systems: R. Tan and M. Shinozuka Note in this respect, Figures 9 and 10 are plotted in logscale while Figure 7 is on the arithmetic scale. Figure 9 clearly indicates that design strategies I and II or the use of M o = 5 and 6 as the design earthquake magnitude again emerge as the optimum strategy. A system expanded from its existing configuration under this strategy is shown in Figure 12, while Figure 11 illustrates the expanded system under design strategy III for comparison. It should be noted that strategies I and II do not always result in the optimum design if the values assumed for the various parameters involved produce higher values of ECF2 and ECF3. Neither of the numerical examples above required overwhelming computational effort. On the contrary, it was possible to use an HP minicomputer system with 21MXE CPU and 20 million byte disk, located at the Department of Civil Engineering and Engineering Mechanics for the numerical analysis. In both cases, execution times were on the order of several minutes to complete the optimization.
Curve B
.200 "7- I00 .~ rtJ
50
20
/
~
CurveA -
I
oinitial cost premium CFI
zxexpected t o tall cost,(ETCF
l
I,II
Ill
IV
V
strategy Figure 9. Initial cost premium and expected total cost for an expanded system.
k=6
~ x~x I
~16
50
~./Curve B Curve A
.,.-, 20 "7
x
~
~ supplynode
'fI[36 '
i '
0 demandnode -- existingpipeline
•
'?
.... newpipeline
~..~A ~:~ 4)
10 i .5
5
(7.6)# /
6
5
-cos, c c_F )
2 / (8.8) g
Figure 11. Optimum expanded system based on strategy III and corresponding pipe sizes.
zx expected incident cost
,
i,I1
i
0 expected repai
/
"
'
iii
(ECF)
I
Iv
[]
strategy Figure 10. Expected repair and incident costs for an expanded system. demand is assumed to be such that the output flow capacity at each demand node must be 25% more than that of the existing system. As previously described, expansion of the existing system is accomplished by looping, for which additional pipelines with diameters k =0.1 ..... 9 are considered. Note that in this case, some of the loops may not be part of the optimum network configuration and therefore may not actually be constructed. The optimizations performed under the five strategies I-Vare summarized in Figures 9 and 10, which indicate that all the costs, CF1, ECF2, ECFa and ETCF are much more sensitive to the strategies to be used than those considered in the first example (see Figures 6 and 7).
~'~ k=6/
o___
supply node
demand node
existing pipeline .... newpipeline
Figure 12. Optimum expanded system based on strategy II and corresponding pipe sizes.
Soil Dynamics and Earthquake Engineering, 1982, Vol. 1, No. 1 37
Optimization o f underground water transmission network systems: R. Tan and M. Shinozuka
CONCLUSION A m e t h o d o l o g y of system o p t i m i z a t i o n has been d e v e l o p e d a n d d e m o n s t r a t e d by means of numerical examples. The o p t i m i z a t i o n was p e r f o r m e d under certain serviceability constraints for the p u r p o s e of identifying the best design strategy. In the present study, the best design strategy represents that strategy which results in the m i n i m u m expected total cost. The total cost consists of the initial cost p r e m i u m of the u n d e r g r o u n d water transmission system, the repair costs associated with p o t e n t i a l seismic d a m a g e to the system a n d the incident (fire) costs representing that p o r t i o n of the indirect fire costs which specifically accrues from the unserviceability of the system due to seismic d a m a g e to itself. The design strategy pertains to the choice of the design e a r t h q u a k e m a g n i t u d e and the decision variables are the pipe diameters of the links (pipelines) of the transmission network. O p t i o n s for the n e t w o r k t o p o l o g y are considered by the possible choice of not c o n s t r u c t i n g (or choice of zero pipe d i a m e t e r for) some of the feasible pipelines. The siting of the source and d e m a n d nodes and routing of links (pipelines) between these nodes are assumed to be p r e d e t e r m i n e d and not part of the o p t i m i z a t i o n p r o b l e m this study addresses itself to. Efficient c o m p u t e r p r o g r a m s were d e v e l o p e d to perform the o p t i m i z a t i o n numerically and two numerical examples were w o r k e d out using the H P m i n i c o m p u t e r system. N o convergence p r o b l e m s arose and the required c o m p u t a t i o n a l time was acceptable even with the use of the m i n i c o m p u t e r system. It was found that the various expected costs are much m o r e sensitive to design strategy when dealing with an o p t i m u m system e x p a n s i o n p r o b l e m rather than when dealing with the o p t i m u m design of a system to be newly constructed. It should be a c k n o w l e d g e d that, possibly, there are m o r e a p p r o p r i a t e constraints o t h e r t h a n e q u a t i o n s (8) and (9). F o r example, the constraints can be such that Pi(majorlMo) is less than a certain specified value. Finally, the same observations as m a d e in reference 1 can be repeated here: (1) The inaccuracies and uncertainties that a c c o m p a n y the various a s s u m p t i o n s used not only in the numerical examples but also in the theoretical d e v e l o p m e n t should be investigated in the future and (2) the scope of the present study should be e x p a n d e d to include the effect of those e a r t h q u a k e s that occur other than a l o n g the San A n d r e a s F a u l t for a m o r e realistic o p t i m i z a t i o n of the system.
F o u n d a t i o n under G r a n t N~o. N S F - P F R - 7 8 - 1 5 0 4 9 with Dr. S. C. Liu as P r o g r a m Manager.
REFERENCES 1
2
3
4
5
6 7 8 9 10
11 12 13 14
15
ACKNOWLEDGEMENT This w o r k was s u p p o r t e d
38
16 by the N a t i o n a l
Science
Soil Dynamics and Earthquake Engineering, 1982, Vol. 1, No. I
Shinozuka, M., Tan, R. and Koike, T., "'Estimation of the Serviceability of Underground Water Transmission Network Systems under Seismic Risk", Technical Report No. NSF-PFR78-15049-CU-6, November 1980; Proceedings of the ASCE Specialty Conjerence on Lifeline Earthquake Engineering, Oakland, California, August 20-21, 1981 Shinozuka, M., Takada, S. and Ishikawa, H., Some Aspects of Seismic Analysis of Underground Lifeline Systems, Journal of Pressure Vessel Technology, ASME, Vol. 101, No. 1, February 1979, pp. 31 34 Shinozuka, M., Takada, S. and lshikawa, H., Seismic Risk Analysis of Underground Lifeline Systems with the Aid of a Damage Probability Matrix, Proceedings of the Fifth Japan Earthquake Engineering Symposium, Tokyo, Japan, November 28 30, 1978, 1423 1432 Shinozuka, M. and Koike, T., Seismic Risk of Underground LifelineSystems Resulting from Fault Movement, Proceedings of the Second U.S. National Conference on Earthquake Engineering, Stanford University, Stanford, California, August 22 24, 1979, 663 672 Shinozuka, M. and Koike, T., Estimation of Structural Strains in Underground Lifeline Pipes, Proceedings of the Lifeline Earthquake Engineering Symposium at the Third U.S. National Congress on Pressure Vessels and Piping, San Francisco, California, June 25 29, 1979, 31 48 Lam, C. F., Discrete Gradient Optimization of Water Systems, Journal of the Hydraulic Division, ASCE, June 1973, 863 872 Watanatada, T., Least Cost Design of Water Distribution Systems, Journal ~ the Hydraulic Division, ASCE, June 1973, 863 872 Cenedese, A. and Mele, P., Optimal Design of Water Distribution Networks, Journal of the Hydraulic Division, ASCE, February 1978, 237 247 Whitman, R. V. et al., Seismic Design Decision Analysis, Journal of the Structural Division, ASCE, May 1975, 1067 1084 Blume,J. A. and Monroe, R. F., The Spectral Matrix Method of Predicting Damage from Ground Motion, John A. Blume and Associates Research Division, Nevada Operations Office of the U.S. Atomic Energy Commission, Las Vegas, Nevada, 1971 Grandori, G. and Benedetti, D. On the Choice of the Acceptable Seismic Risk, International Journal oJ Earthquake Engineering and Structural Dynamics, Vol. 2, Nol 1, 1973, 3 10 kiu, S. C. and Neghabat, F., A Cost Optimization Model for Seismic Design of Structures, The Bell System TechnicalJournal, Vol. 51, No. 10, 1972, 2209 2225 Shah, H. C. and Vagliente, V. N., Forecasting the Risk Inherent in Earthquake Resistant Design, Proceedings of the International ConJerence on Microzonation, Vol. 11, 1972, 693 718 Whitman, R. V. and Hein, K. H., Damage Probability for a Water Distribution System, Proceedingsof the ASCE TCLEE Specialty Conference on Lifeline Earthquake Engineering, August 31Y31, 1977, 410-423 Meredith, D. D. et al., Design and Planning oj Engineering Systems, Prentice-Hall Inc., New Jersey, 1973 Himmelblau, D. M., Applied Nonlinear Programming, McGrawHill Inc.. New York, 1972
The purpose of this paper is to establish procedures for the optimum design of underground water transmission systems to be newly constructed as well as for the optimum improvement of existing systems on the basis of seismic risk and cost-effectiveness considerations. As an example, a realistic water transmission system will be analyzed and optimized. Procedures for optimizing the total cost are developed in which the pipe diameters and the network topology are parameters which can be varied to achieve a minimum total cost under certain realistic constraints. The total cost includes the initial cost premium, repair costs and incident (fire) costs. The optimization algorithm employed in this study is derived from the discrete gradient method.
INTRODUCTION In a recent paper 1, Shinozuka et al developed a methodology for the assessment of the seismic risk which underground water transmission systems in seismically active areas are subjected to, in terms of the degree of possible unserviceability resulting from destructive earthquakes. In that study, it was postulated that a water transmission system was serviceable when its firefighting capabilities (measured in terms of water pressure and flow rate at specified nodes) remained intact immediately following an earthquake. With the aid of techniques developed by Shinozuka et al in previous papers 2 - 5, the probabilities of link failure were evaluated and Monte Carlo techniques were used to generate a sample of simulated states of damage for the water transmission network. The Newton-Raphson method was then utilized to perform a flow analysis on each of these damaged networks and, at the same time, to determine whether or not it still remained serviceable. A numerical example was presented to demonstrate the probabilities of system unserviceability estimated under a set of assumed parameter values deemed reasonable. However, the construction or improvement of water transmission systems usually involves a large capital outlay and continuous expenditures for maintenance and repairs that are only balanced by the various benefits the systems can generate. Therefore, it is necessary to perform a "seismic design decision analysis" on such systems in order to examine the advisability of alternative design decisions on the basis of the cost-effectiveness of the feasible strategies for the design, construction and improvement of those systems subjected to seismic risk. The purpose of this paper is to develop a design and improvement optimization methodology within the framework of such a seismic design decision analysis. Achieving this purpose is considerably difficult due to the complex operational as well as structural functions the network system must perform to satisfy the prescribed serviceability criteria, e.g., water flow rate and pressure
requirements. The difficulty is compounded even further because of the wide range of choices available for such design and improvement strategies involving such inputs as the pipe size, pipe material, joint structure, buried depth, material and construction method of back fill, pipe routing, seismic magnitude to be used in the design, etc., under competing constraints on the system's reliability and economics. Some recent papers by Lam 6, Watanetada 7 and Cenedese and Mele a have considered such optimization problems, specifically those dealing with water transmission and distribution systems without, however, taking into account the potential seismic destruction of the system. At the same time, more general aspects of the optimum design decision analysis for engineering systems in seismic environments with strong socioeconomic implications have also attracted increasingly serious attention among engineers; for example, Whitman et al 9, Blume and Monroe 1°, Grandori and Benedetti ~1, Liu and Neghabat 12 and Shah and Vagliente ~3. Not much work in this direction with respect to underground water transmission network systems has been pursued, however. In the present study, the optimum design of a water transmission system is derived on the basis of a design decision analysis involving cost-benefit considerations so as to balance the initial cost of the design, construction and improvement of the system and the future losses that could accrue from earthquake-induced malfunctions of the same system. Such losses are usually classified into two categories: the direct loss, somewhat easier to evaluate, results from such post-earthquake activities (necessitated by the unserviceability) as emergency and permanent repairs of the seismic damage sustained by the system, emergency water delivery, etc. On the other hand, the indirect loss consists of inconveniences suffered by residential users, damage due to postearthquake fires, industrial loss, etc., pertaining to the malfunction or unserviceability of the system. The indirect loss is therefore much more difficult to assess.
0261 7277/82/01(X)3(~ 0952.00 © 1982 CML Publications
30
Soil Dynamics and Earthquake Engineering, 1982, Vol. 1, No. 1
Optimization of underground water transmission network systems: R. Tan and M. Shinozuka The specific optimization methodology to be developed in the sequel is based on an analysis that examines the sum total of the initial cost and the (indirect as well as direct) future losses as a function of the design or improvement strategy and recommends a strategy which will achieve the optimum cost. The step-by-step procedure of such an analysis can be delineated as follows: (1) a mathematical model of the water transmission network is constructed. (2) A cost or objective function is introduced as a function of the pipe length and diameter and the expected seismic losses. (3) The discrete gradient method is employed to find the solution that minimizes the cost function under appropriate constraining conditions involving the system performance criteria.
NETWORK TOPOLOGY In dealing with a water transmission system to be newly constructed, the present study assumes that the locations of the source nodes (water supply stations) and demand nodes are predetermined. Therefore, finding their optimum locations is not part of the analysis to be performed here. However, it is acknowledged that the siting problem associated with these source and demand nodes, while not specifically considered here, is essentially that of overall system optimization, and its solution requires careful consideration not only of the spatial as well as temporal distributions of the water demands for domestic, industrial and other consumption, but also of such environmental constraints as the topographical and geological conditions of the general area in which the transmission system is to be constructed. With respect to the links (or pipelines) that will connect these source and demand nodes, their routing also requires intelligent techno-economic judgment in order to select only those options that are viable and feasible from the viewpoint of the construction costs. Those routes that would be in direct conflict with other major structures, existing or planned, and that require passage through private property and/or regions under unfavourable topographical, geological and soil conditions, should be considered unfeasible. Those routes that would produce a redundancy in the transmission network only at an obviously disproportionate cost should also be considered unfeasible. Thus, the decision on the routing of links cannot be made as arbitrarily as might appear. For this reason, the present study performs an optimization analysis on a network for which not only the nodes but also all possible routes for the links are prescribed. In fact, Figure l(a) illustrates such a network in which a dashed line connecting two nodes implies a feasible pipeline construction between them. If there is no dashed line indicated between two nodes (e.g., between nodes 3 and 5), it means that the construction of a pipeline between these two nodes has been ruled out on the basis of infeasibility (but not of optimality). As mentioned above, the infeasibility may be due to (a) interference with other major structures, (b) unfavourable topographical, geological and soil conditions and (c) obvious uneconomical redundancy. Under these circumstances, the diameter of the pipes (assumed to be uniform at least within each link) becomes the only parameter that can be varied to achieve the optimum design of the system. Indeed, the optimization method to be described later will indicate which possible
routes should or should not be actually constructed for the optimum design. It will also indicate the optimum diameter of the pipe in each of the links to be constructed. The total number of links, therefore, may be decreased but never increased as a result of such an optimization analysis. For example, the analysis may find that an optimum design is achieved if all the feasible links in Figure l(a) except for links 1 3, 6-8 and 5-8 are constructed, thus resulting in an optimum network as shown in Figure l(b). It is not uncommon to find that an optimally designed water transmission system, e.g., Figure l(b), becomes inadequate after a period of time due to the growth of population and the increased industrial activities within the community served by the system. In order to meet these new demands, the system must be expanded by either replacing the old pipelines with those possessing a larger capacity or adding new pipelines to the network system along feasible routes that were not used for the optimum design: routes 5-8, 6-8 and 1-3 in Figure l(a). With respect to the replacement of old pipelines, however, it is often more economical to construct a new pipeline parallel to an existing one. This practice is known as "looping" and it obviously results in increased link capacity. Figure 2 shows a possible expansion of the network in Figure l(b) in response to increased demands at nodes 3 and 5. The expansion reconsiders not only the possible use of those feasible routes that were not part of the initial optimum design (routes 5 8, 6 8 and 1-3) but also the looping of all existing links. Hence, in Figure 2, solid lines indicate existing pipelines, while dashed lines possible new pipelines. The optimization analysis will then determine which new pipelines should or sl~ould not be constructed.
9
9 I
I
',
I
-
~5
,~ I
\
I
I
I
I
"-6 .....
-6,4
\
7(S.
~l
i
8
8
(a)
I0
10
(b) source node
0
demand node link feasible link
Figure 1. Newly constructedsystem.
Soil Dynamics and Earthquake Engineering, 1982, Vol. 1, No. 1 31
Optimization of underground water transmission network systems: R. Tan and M. Shinozuka
9
92 3 7
•
% node with new
I0
demand
A
source node
0
demand node existing link
---
Fioure 2.
feasible new l i n k
Expanded system.
COST F U N C T I O N S The initial cost function CFx, which includes the cost of the pipes themselves and the cost of construction, represents the initial cost premium. The cost CFou per unit length of the pipeline between nodes i a n d j depends on the pipe diameter D~jk (uniform throughout its length Lij) and other factors unique to link ij (e.g., soil conditions along the route). The subscript k herein refers to the pipe size number. Using a generally accepted monomial formula, the cost CFo~j is then approximated by CFo~j = w~jD~jk, where 7 is a nondimensional constant between 1.0 and 2.0 and w~j is the so-called cost index for the pipeline ij which reflects the constrqction as well as pipe costs. Then, the initial cost premium can be expressed by
NONO CFx = Y, ~', wijD~jkLijtij i j
i , j = l , 2, .. ND i>j "'
(1)
where L u = the length of link ij connecting nodes i and j, ND = the total number of demand and source nodes and tij = the topological index: tij = 0 if link ij is unfeasible and tij = 1 if link ij is feasible. For example, in Figure l(a), t35 =0, t24 =0, t16 = I, t58-- l, etc. The present optimization analysis permits the pipe diameter to vary within a set of discrete values: Dij o =0, Dijt = 101.6 cm (40'3, Dij2 = 114.3 c m (45"),..., Dij 9 = 203.3 cm (80"). If the analysis indicates that t~j = 1 but D~jk= 0 (or k = 0) for an optimum design, it means that route ij is feasible but the optimality requires that it not be constructed. The thickness of the pipe is assumed to be 16 mm, regardless of the diameter size. As mentioned earlier, in addition to the initial cost premium CF1, the direct and indirect losses resulting from earthquakes must enter the cost-benefit formulation. With respect to direct loss, only the repair cost will be considered in the present study for the sake of simplicity.
32
As to the indirect cost, only the loss due to postearthquake fire will be considered in order to be consistent with the serviceability criteria postulated and investigated 1. The disasters associated with the 1906 San Francisco earthquake and the 1923 Tokyo earthquake warrant such a study dealing with the post-earthquake fire-fighting capability of water transmission systems. Indeed, in the case of the San Francisco earthquake, the damage directly attributable to ground shaking was estimated at $24 million while the loss due to the fire caused by the earthquake was $500 million (in terms of 1906 dollars). Also, the Tokyo earthquake claimed 143,000 human lives primarily due to the fire that immediately followed the earthquake. The optimization procedure to be used here is based on the expected total cost to be estimated with the aid of a damage probability matrix for each link and an unserviceability probability matrix for each node. Table 1 shows the damage probability matrix for link ij and its nm element indicates P[Lijk,lm], the probabilities of link ij in the damage state n, given that the earthquake magnitude M = m. Note that the event L~jkl = the state of link ij in major damage, or equivalently, failure of link ij, L~k3 = the state of link ij in minor damage, or survival of link ij and L~jk2 = the state of link ij in moderate damage. These probabilities were obtained from reference 1. Similarly, the unserviceability probability matrix at node i is shown in Table 2. Its n m element indicates P~(n[m) where Pi(llm), Pi(2lm) and Pi(3lm) are written for Pi(majorlm), Pi(moderatelm) and Pi(minorlm), respectively. Each element refers to the probability of minor, moderate or major unserviceability associated with node i, given that the earthquake magnitude M = m. As mentioned ~, a state of major unserviceability corresponds to the condition that at least either the water pressure or the flow rate is less than the respective minimum requirements (Emi n for pressure and Qmin for flow rate) for firefighting. A state of moderate unserviceability represents the condition under which both the pressure and flow rate satisfy the minimum requirements but do not simultaneously exceed the design allowable amounts associated with the undamaged original network (Er.od for pressure and Qmod for flow rate). A state of minor
Table 1. Damage probability matrixfor link ij Damage states, n 1. Major 2. Moderate 3. Minor
Central damage ratio
CDR1 CDR2
Earthquake magnitude, m 5
6
7
8
9
P[LisR. Im]
CDR3
Table 2. Unserviceabilityprobability matrix for node i Earthquake magnitude, m Unserviceability U n s e r v i c e a b i l i t y 5 6 7 8 9 states, n cost ratio 1. Major 2. Moderate 3. Minor
Soil Dynamics and Earthquake Engineering, 1982, Vol. 1, No. 1
UCR~ UCR2 UCR 3
Pi(nqm)
Optimization of underground water transmission network systems: R. Tan and M. Shinozuka unserviceability is then defined as that in which the pressure and flow rate exceed Emod and Qmod, respectively, and therefore implies a state of normal serviceability. These probabilities were also obtained from reference 1. The second column of Table 1 lists the central damage ratio CDR, as a representative value of the ratio of the repair cost of the pipeline (between nodes i and j) in the damage state n to its replacement costs 9'14. Usually, primarily due to the paucity of relevant information, reasonable values are selected for these ratios essentially on the basis of engineering judgment. Utilizing the central damage ratios thus introduced, the expected value of the repair cost CF2 for a period of T years can be written in approximation as
ND ND 3 ECF2 = ~ Z wijDi]kLi/ij x T Z CDR. x P[Lijk,] x DC i
j
n=l
i>j
(2)
capability of the water transmission system would, on the average, be greater than those under that of an intact system. As far as the indirect losses are concerned, it is this incremental component of the fire losses that this study addresses itself to. Therefore, the second column in Table 2 lists the unserviceability cost ratio, UCR., which reflects such an incremental component of fire losses and is defined below: Consider a representative value of the ratio of the fire losses that would be incurred if the water transmission system is in the damage state n immediately following an earthquake to the fire losses if the system remains intact. Subtract unity from this ratio. This then is the unserviceability cost ratio pertaining to the fire losses. This ratio is perhaps even more difficult to estimate than the central damage ratio. For the purpose of numerical analysis, however, it is assumed that U C R 3 = 0, U C R 2 = 2 and UCRI=4. The expected value of the indirect losses C F 3 due to fire for a period of T years can then be written as
with
N' 3 E C F 3 = T E E EFCi × UCR, x P~(n)× DC
/5[L,~k,] = v ~ P[ giik,lm] {F M(m+ 0.5) -- F M(m-- 0.5)} (3)
(5)
i=ln=l
m
where w~jD~jkLgjhas been interpreted as the replacement cost, v is the annual rate of earthquake occurrence, DC is a discount factor, and FM(') is the probability distribution function of M: FM(m) = { 1 -- [ e x p -- fl'(m -- mL) ] }/{ 1 - e x p [ - fl'(m u - m L)] } (4)
where m L is the lower bound of the magnitude value of engineering significance ( = 4.0 in the present study), while mu is the largest magnitude value expected from the earthquake, and fl' is a site-dependent constant. In this study, mu=9. Thus, the product v{FM(m+O.5)-FM(m -0.5)} indicates the annual occurrence probability of an earthquake with a magnitude between m - 0 . 5 and m + 0.5. The quantity P[Li~k,] in equation (2) represents the expected value of the annual failure probabilities of the pipeline between nodes i and j in the damage state n and
with
P,(n) = v ~ P,(n[m){FM(m+ 0.5) -- FM(m-- 0.5)}
(6)
m
where N' is the total number of demand nodes. Also, in equation (5), EFCi represents the expected fire incident costs for the area that is covered by node i as protection against fire when the water transmission system survives an earthquake without damage. As such, EFCi depends on the characteristics of each region represented by the type of buildings and structures, residential, commercial or industrial, to be found therein. Again, it is rather difficult to estimate such a quantity without recourse to subjective judgment. Finally, the expected total cost E T C F can be constructed as
ECTF = CF1 + ECF2 + ECF3
3
CDR, x P[Lijkn ] is the corresponding expected value n=l
of the annual damage ratio. There is always the possibility of a post-earthquake fire and once such a fire occurs, it will inflict a certain amount of loss upon the community in general. The amounts of such losses (categorized as indirect losses in the present study) depend on a number of inherently probabilistic factors: for example, (a) exactly when, where and how it occurs and under what weather conditions, (b) to what extent the community fire-fighting force remains effective in terms of the availability of manpower and fire engines and in view of the accessibility of the fire engines to those areas of conflagration. Such availability and accessibility can be seriously hampered by the possible destruction of the fire engine stations, bridges, highways, etc. and by the blockage of streets by collapsed structures and debris. And of more relevance to the present study, (c) how serviceable the water transmission system remains immediately after the earthquake in terms of its firefighting capability. Particularly with respect to item (c) above, the losses due to fire under the degraded conditions of fire-fightlng
NONO / + = x DC) = ~i ~j wijD~'jkLijtlj~ 1 T ~ , CDR,× "[Lijkn]
N' 3 + T ~ ~, EfC, x UCR. x Pi(n) x DC i=1
i>j
(7)
n=l
where P[Li~R,] and Pi(n) are given by equations (3) and (6), respectively. The preceding formulation of the expected costs, particularly in terms of equations (2), (5) and (7), applies to water transmission systems to be newly constructed. However, the same equations can be employed for expanding a system, with the following modifications: (a) Only the initial cost premium of the new pipelines is considered for CF1 and (b) ECF2 includes only those repair costs for the new pipelines. Several economic criteria such as the net present worth, annual equivalent value and benefit-cost ratios 15 can be used by designers and decision makers to rank alternative design and improvement strategies and to select the best among them, provided that all the costs and benefits associated with each strategy can be estimated. In the
Soil Dynamics and Earthquake Engineering, 1982, Vol. 1, No. 1 33
Optimization of underground water transmission network systems: R. Tan and M. Shinozuka 2B
~2
¢xl
2 "7.
I
-----~direction
of steepest
descent at point I
• /
~2A
2Cj
g "7,
it_
Decision Variable I
Figure 3. Steepest descent in discrete decision variable space. present study, however, the expected total cost function E T C F in equation (7) is considered the objective function to be minimized for optimization with respect to the decision variables D~jk under the following constraints at all the nodes:
Ei(Mo) >1Emin
i = 1, 2..... N'
(8)
Oir(MD) >~Qmin
i = 1, 2..... N'
(9)
where E,i(Mo) and Qi,(Mo) are, respectively, the conditional expectations of El, the water head at node i, and Q~,, the rate at which the water is extracted from node i, given that the earthquake magnitude is equal to the design magnitude Mt~ These conditional expectations can be obtained from reference 1. In the present study, selecting a specific value for the design earthquake magnitude M o from among the various possible alternatives (say, M o = 5, 6, 7, 7.5 and 8) is considered, for simplicity, to represent the entire design strategy, although in reality the design strategy implies a set of feasible choices based on a number of pertinent design parameters. Therefore, the constraining conditions given by equations (8) and (9) are of particular importance in view of the fact that it is only through these conditions that the design earthquake magnitude is reflected in the design or in the choice of optimum pipe diameters. DISCRETE GRADIENT METHOD The gradient method is used to minimize the objective function E T C F in equation (7) under the constraints of equations (8) and (9). As is well known, this method finds the path of steepest descent along which the numerical iteration can proceed to eventually locate a local minimum 16. Since the present decision variables D~jk and hence the objective function are discrete, it is not possible to evaluate in the usual sense the gradient of the objective function from which the direction of steepest descent can be found. However, this difficulty is not a serious one and can be circumvented by the use of the interpolated objective function which is defined over a continuous range of pipe diameter values. Further difficulties lie in the fact that (a) the objective function E T C F in equation (7) involves the probabilities of link failure and unserviceability in various states of damage which are extremely complicated functions of the decision variables D~jk and (b) the direction of steepest descent will not necessarily pass through a feasible point in the decision variable space due to the discreteness of the decision variables. Figure 3 schematically illustrates this
34
for the case involving two decision variables. To overcome difficulty (a) above, the direction of steepest descent for the initial cost premium CF1 given by equation (1) will be used in approximation as the direction of steepest descent associated with the expected total cost function E T C F in the iterative procedure of finding the local minimum. Such an approximation will drastically reduce the amount of numerical effort to be otherwise expended in such an optimization procedure since, in this way, the direction of steepest descent can be found analytically. Obviously, the approximation will produce a reasonable result if the behaviour of the sum of the expected repair costs ECF2 and the expected incident (fire) costs ECFa, with respect to variation in the decision variables, is similar to that of the initial cost premium. This is numerically confirmed to be the case in examples considered later. Difficulty (b) above can be avoided by following the suggestion made by Lam 6 on the basis of empirical results. The cosine of the angle 0ij between the direction of steepest descent and the axis of the decision variables Dijk can be shown to be cos Oil= -
/~
,~,ij,lijk ,~ij,ij
i >j
(10)
4~i ~ ('wijD[~klLutu)2 Lam's procedure utilizes these cosine values in the following fashion: Suppose, at a particular cycle of iteration, that a certain number of pipelines have cosine values larger than 0.5; cos0ij~>0.5. In this case, (a) the diameters of these pipelines will be decreased from Dijk to Dij~k- 1,- These reduced values of pipe diameters, together with the ones unchanged, represent a new decision point along the path of steepest descent. (b) A check will be made to see if this new decision point is a feasible point in terms of satisfying the constraining conditions. (c) If it is a feasible point, the path of steepest descent will be extended further from this point and the process of optimization will continue. (d) If it is not, the diameter of the pipeline with the largest cosine value will be reduced from D~jk to Dij(k 1, thus establishing a new decision point. (e) A check is made to examine if it is in the feasible domain. (f) If it is, the path of steepest descent is extended and the iteration proceeds. (g) If it is not, the diameter of the pipeline with the second largest cosine value will be reduced, by one size, the feasibility will be checked, and the procedure will be repeated until a new feasible decision point is found. It may happen, however, that no new decision point can be found in the feasible domain by adjusting the diameter of the pipelines with cos 01j >~0.5. If this happens, then (h) the reduction (by one size) of the diameter of those pipelines with cos 0~j < 0.5 will be considered one by one in such an order that the pipeline with the larger cosine value will be considered before one with a smaller cosine value. (i) If none of the pipelines involved produces, upon reduction (by one size) of its diameter, a new decision point in the feasible domain, the current decision point represents the local (and possible global) optimum design. (j) The above procedure results, in general, in a different local minimum, depending on the initial decision point of the initial values that must be assigned to the decision variables for the purpose of starting the iteration. If such initial values produce cos0~j which are smaller than 0.5 for all the pipelines involved, then the procedure should begin at step (h).
Soil Dynamics and Earthquake Engineering, 1982, Vol. 1, No. 1
Optimization of underground water transmission network systems: R. Tan and M. Shinozuka Table 3. Scheme for findin9 the 91obal minimum initial design
links "--....point with
~
larger sizes
1"
8
9
0
=
,-
E ~
2
3
4
0
0
5
6
7
smaller sizes
0 +
+
0 + +
intermediate sizes
+
+
+
* indicates iteration + indicates -- indicates 0 indicates
0
0
+
10
8
the use of the largest available size for each link starting the increasing the pipe size by one level; Dqk~Dq(k+ ~j decreasing the pipe size by one level; Dok~Di~(k 17 no change
While the procedure described above certainly yields a local minimum, there is no practical method that can be used to locate the decision point corresponding to the global minimum for the present optimization problem. Therefore, the present study uses a standard technique in which the global minimum is estimated as the minimum of a number of local minima resulting from iterations initiated at different sets of initial decision points. In the present study, these initial decision points are not specified at random in the decision variable space as is often done when solving this type of optimization problem, but by means of a systematic scheme as illustrated in Table 3. This scheme economically generates the initial decision points, thereby increasing the possibility that the minimum of the resulting local minima will indeed be the global minimum. The second column of Table 3 corresponds to an optimization process starting at the first initial decision point which represents the use of the maximum available pipe size for all the pipelines. This first initial decision point will result in the first minimum value of the objective function and the corresponding first optimum design. The third column of Table 3 then indicates an optimization process starting at the second initial decision point which is obtained by adjusting the first optimum design. This adjustment consists of increasing, by one size, the diameter of the intermediately sized pipelines (intermediate in diameter size) in the first optimum design. This second initial decision point will lead to the second minimum value of the objective function and the corresponding second optimum design, each of which may or may not be the same as the corresponding quantities associated with the first decision point. If the second minimum value of the objective function is smaller than the first minimum, the third initial decision point is obtained from the second optimum design by increasing (by one size) the diameter of the smaller sized pipelines in the latter. If not, the third initial decision point is derived from the first optimum design by increasing (by one size) the diameter of the smaller sized pipelines in the first optimum design. This procedure continues in accordance with Table 3 which indicates how the new initial design points can be systematically generated by adjusting the optimum design points already obtained. The numerical examples that follow will show that the number of different initial decision points needed to track down the global minimum is usually less than ten. N U M E R I C A L EXAMPLES Two examples are presented: In the first example, the optimum design of a water transmission system is
considered, while in the second, the optimum expansion of the transmission system optimized in the first example is demonstrated. In fact, the water transmission network to be considered herein is a simplified version of the Los Angeles system and consits of three (3) supply stations, ten (10) demand nodes and eighteen (18) links, as shown in Figure 4. The physical data to be used in these examples are the same as those used in reference 1. Those commercially available sizes for pipe diameters are assumed to be those corresponding to k = 1, 2. . . . . 9 in the first example. The diameters of the pipelines to be added to the existing system in the second example are those corresponding to k = 0, 1, 2 ..... 9. This implies that in the first example, all the links shown in Figure 4 must be constructed and thus the optimum design maintains the same topological characteristics as those in Figure 4 since k---0 is not included an an option in this example.
Optimization of a system to be newly constructed As mentioned above, the water transmission system depicted by in Figure 4 is considered. By assumption, all the links shown are feasible and must be constructed (k :~ 0). Also, it is assumed that 7 = 2.0, wo = 1 and the five design strategies, l-V, respectively, represent the design earthquake magnitudes MD= 5, 6, 7, 7.5 and 8. Figure 5 indicates the variation of the optimum initial cost premiums, under strategies I (curve A) and III (curve B), computed for different starting decision points under the scheme illustrated by Table 3. The minimum values among these local optima are observed for starting decision points 6, 8, 9, 11 and 12 under Strategy I. These values are identical within a small range of computational error and are considered as the global minima under this strategy. Similarly, under Strategy lII, the global minimum is estimated as the local optima associated with starting decision points 9 and 12. For ease of comparison, these values are plotted in Fig. 5 in terms of a percentage of the global minimum value under Strategy I. Also in Figure 5, the numbers in parentheses indicate those of the iterations needed to reach the corresponding optimum values. Both curves A and B suggest that the local minimum, which will eventually be taken as the global
13
1 supply station demand node 2
3
12[
11
Figure 4.
Water distribution system of Los Angeles.
Soil Dynamics and Earthquake Engineering, 1982, Vol. 1, No. 1 35
Optimization of underground water transmission network systems: R. Tan and M. Shinozuka (24)
(]75) (i81
(27) -~C~ve
B for Strategy III (19)
(23)
114 112(83)
g g T~
\-
Ii0~(23
)
I 0 8 ~ 106
[numbers in parentheses indicate the number of iterations for convergence]
required
(26) (23)
o/L
~
(23)
104
O) Curve A for
2
102 Strategy I
(2)
(24)
conservative strategy is adopted, the repair costs increase but their relative percentage values decrease since the initial cost premiums increase more rapidly. As shown in Figure 7, however, the incident fire costs respond to conservative strategies much more favourably by exhibiting a sharper decrease per larger initial cost investment. The expected total costs are indicated by curve B in Figure 6 which concludes that design strategies I and II share optimality. The corresponding diameter sizes of the pipelines are shown in Figure 8 in terms of k values.
(15) 2'
3
4
5
6
~
8
9
1'0
1'I
1'2
13
starting points for iteration
Figure 5. Initialcostpremiumford~erentstartingpoints.
Optimum expansion of an existing system The system optimized in the preceding example and shown in Figure 8 is considered the existing system upon which an increased demand has been imposed. The new
o initial cost premium(CF1) expected total cost 300
(ETCF) .16
°r(,'-,,i
25O
oexpected repair cost n expected incident(ECF2)/~ cost (ECF3/ 1
2OO
+,
-14 C u r v e B ~ ~ /
150
CuI
.12
> ,m'-
m-'--
Curve A
100 I
I,II III strategy
I
IV
0"I
I
V
I
[(I+i) T - 1]/[Ti(1 +i)~
I
I,II III strategy
Initial cost premium and expected total costs.
DC=
(8.2)
(8.7)
8
minimum appears before the tenth initial decision point is tried. The global minima of CF1 associated with the five design strategies I V are plotted in Figure 6 (curve A) while the corresponding expected repair costs ECF2 and the expected incident (fire) costs ECFa are shown in Figure 7 for the same design strategies. The expected repair and incident costs are computed under the assumptions that (a) CDR 1,2,3 =0.9, 0.4, 0.0, (b) UCRI.2.3 = 4, 2, 0, (c) the design life of the system = fifty (50) years, (d) the annual rate of earthquake occurrence v = 3, (e) the expected fire incident costs EFCI for the area that is covered by demand node i as protection against fire in an undamaged system are 1.33~o, 5.32~o, 5.32~o, 1.33°~o,2.66~ , 3.990, 6.65~o, 3.99~o, 6.65~0, 2.66% of the initial cost premium CFx of the system, respectively, for i = 1 - 10, (f) the discount factor is given as
I
I
IV
V
Figure 7. Expected repair and incident costs. k=8
I-l supply node 6
1
J
(11)
with the discount rate i=8~o. The numbers in parentheses in Figure 7 indicate percentage figures of the repair costs with respect to the initial cost premiums under the same design strategies. When a more
36
/~ \
-10 I
Figure 6.
(6"4)
Figure 8.
Soil Dynamics and Earthquake Engineering, 1982, Vol. 1, No. 1
Optimum pipe sizes.
0
demand node
Optimization of underground water transmission network systems: R. Tan and M. Shinozuka Note in this respect, Figures 9 and 10 are plotted in logscale while Figure 7 is on the arithmetic scale. Figure 9 clearly indicates that design strategies I and II or the use of M o = 5 and 6 as the design earthquake magnitude again emerge as the optimum strategy. A system expanded from its existing configuration under this strategy is shown in Figure 12, while Figure 11 illustrates the expanded system under design strategy III for comparison. It should be noted that strategies I and II do not always result in the optimum design if the values assumed for the various parameters involved produce higher values of ECF2 and ECF3. Neither of the numerical examples above required overwhelming computational effort. On the contrary, it was possible to use an HP minicomputer system with 21MXE CPU and 20 million byte disk, located at the Department of Civil Engineering and Engineering Mechanics for the numerical analysis. In both cases, execution times were on the order of several minutes to complete the optimization.
Curve B
.200 "7- I00 .~ rtJ
50
20
/
~
CurveA -
I
oinitial cost premium CFI
zxexpected t o tall cost,(ETCF
l
I,II
Ill
IV
V
strategy Figure 9. Initial cost premium and expected total cost for an expanded system.
k=6
~ x~x I
~16
50
~./Curve B Curve A
.,.-, 20 "7
x
~
~ supplynode
'fI[36 '
i '
0 demandnode -- existingpipeline
•
'?
.... newpipeline
~..~A ~:~ 4)
10 i .5
5
(7.6)# /
6
5
-cos, c c_F )
2 / (8.8) g
Figure 11. Optimum expanded system based on strategy III and corresponding pipe sizes.
zx expected incident cost
,
i,I1
i
0 expected repai
/
"
'
iii
(ECF)
I
Iv
[]
strategy Figure 10. Expected repair and incident costs for an expanded system. demand is assumed to be such that the output flow capacity at each demand node must be 25% more than that of the existing system. As previously described, expansion of the existing system is accomplished by looping, for which additional pipelines with diameters k =0.1 ..... 9 are considered. Note that in this case, some of the loops may not be part of the optimum network configuration and therefore may not actually be constructed. The optimizations performed under the five strategies I-Vare summarized in Figures 9 and 10, which indicate that all the costs, CF1, ECF2, ECFa and ETCF are much more sensitive to the strategies to be used than those considered in the first example (see Figures 6 and 7).
~'~ k=6/
o___
supply node
demand node
existing pipeline .... newpipeline
Figure 12. Optimum expanded system based on strategy II and corresponding pipe sizes.
Soil Dynamics and Earthquake Engineering, 1982, Vol. 1, No. 1 37
Optimization o f underground water transmission network systems: R. Tan and M. Shinozuka
CONCLUSION A m e t h o d o l o g y of system o p t i m i z a t i o n has been d e v e l o p e d a n d d e m o n s t r a t e d by means of numerical examples. The o p t i m i z a t i o n was p e r f o r m e d under certain serviceability constraints for the p u r p o s e of identifying the best design strategy. In the present study, the best design strategy represents that strategy which results in the m i n i m u m expected total cost. The total cost consists of the initial cost p r e m i u m of the u n d e r g r o u n d water transmission system, the repair costs associated with p o t e n t i a l seismic d a m a g e to the system a n d the incident (fire) costs representing that p o r t i o n of the indirect fire costs which specifically accrues from the unserviceability of the system due to seismic d a m a g e to itself. The design strategy pertains to the choice of the design e a r t h q u a k e m a g n i t u d e and the decision variables are the pipe diameters of the links (pipelines) of the transmission network. O p t i o n s for the n e t w o r k t o p o l o g y are considered by the possible choice of not c o n s t r u c t i n g (or choice of zero pipe d i a m e t e r for) some of the feasible pipelines. The siting of the source and d e m a n d nodes and routing of links (pipelines) between these nodes are assumed to be p r e d e t e r m i n e d and not part of the o p t i m i z a t i o n p r o b l e m this study addresses itself to. Efficient c o m p u t e r p r o g r a m s were d e v e l o p e d to perform the o p t i m i z a t i o n numerically and two numerical examples were w o r k e d out using the H P m i n i c o m p u t e r system. N o convergence p r o b l e m s arose and the required c o m p u t a t i o n a l time was acceptable even with the use of the m i n i c o m p u t e r system. It was found that the various expected costs are much m o r e sensitive to design strategy when dealing with an o p t i m u m system e x p a n s i o n p r o b l e m rather than when dealing with the o p t i m u m design of a system to be newly constructed. It should be a c k n o w l e d g e d that, possibly, there are m o r e a p p r o p r i a t e constraints o t h e r t h a n e q u a t i o n s (8) and (9). F o r example, the constraints can be such that Pi(majorlMo) is less than a certain specified value. Finally, the same observations as m a d e in reference 1 can be repeated here: (1) The inaccuracies and uncertainties that a c c o m p a n y the various a s s u m p t i o n s used not only in the numerical examples but also in the theoretical d e v e l o p m e n t should be investigated in the future and (2) the scope of the present study should be e x p a n d e d to include the effect of those e a r t h q u a k e s that occur other than a l o n g the San A n d r e a s F a u l t for a m o r e realistic o p t i m i z a t i o n of the system.
F o u n d a t i o n under G r a n t N~o. N S F - P F R - 7 8 - 1 5 0 4 9 with Dr. S. C. Liu as P r o g r a m Manager.
REFERENCES 1
2
3
4
5
6 7 8 9 10
11 12 13 14
15
ACKNOWLEDGEMENT This w o r k was s u p p o r t e d
38
16 by the N a t i o n a l
Science
Soil Dynamics and Earthquake Engineering, 1982, Vol. 1, No. I
Shinozuka, M., Tan, R. and Koike, T., "'Estimation of the Serviceability of Underground Water Transmission Network Systems under Seismic Risk", Technical Report No. NSF-PFR78-15049-CU-6, November 1980; Proceedings of the ASCE Specialty Conjerence on Lifeline Earthquake Engineering, Oakland, California, August 20-21, 1981 Shinozuka, M., Takada, S. and Ishikawa, H., Some Aspects of Seismic Analysis of Underground Lifeline Systems, Journal of Pressure Vessel Technology, ASME, Vol. 101, No. 1, February 1979, pp. 31 34 Shinozuka, M., Takada, S. and lshikawa, H., Seismic Risk Analysis of Underground Lifeline Systems with the Aid of a Damage Probability Matrix, Proceedings of the Fifth Japan Earthquake Engineering Symposium, Tokyo, Japan, November 28 30, 1978, 1423 1432 Shinozuka, M. and Koike, T., Seismic Risk of Underground LifelineSystems Resulting from Fault Movement, Proceedings of the Second U.S. National Conference on Earthquake Engineering, Stanford University, Stanford, California, August 22 24, 1979, 663 672 Shinozuka, M. and Koike, T., Estimation of Structural Strains in Underground Lifeline Pipes, Proceedings of the Lifeline Earthquake Engineering Symposium at the Third U.S. National Congress on Pressure Vessels and Piping, San Francisco, California, June 25 29, 1979, 31 48 Lam, C. F., Discrete Gradient Optimization of Water Systems, Journal of the Hydraulic Division, ASCE, June 1973, 863 872 Watanatada, T., Least Cost Design of Water Distribution Systems, Journal ~ the Hydraulic Division, ASCE, June 1973, 863 872 Cenedese, A. and Mele, P., Optimal Design of Water Distribution Networks, Journal of the Hydraulic Division, ASCE, February 1978, 237 247 Whitman, R. V. et al., Seismic Design Decision Analysis, Journal of the Structural Division, ASCE, May 1975, 1067 1084 Blume,J. A. and Monroe, R. F., The Spectral Matrix Method of Predicting Damage from Ground Motion, John A. Blume and Associates Research Division, Nevada Operations Office of the U.S. Atomic Energy Commission, Las Vegas, Nevada, 1971 Grandori, G. and Benedetti, D. On the Choice of the Acceptable Seismic Risk, International Journal oJ Earthquake Engineering and Structural Dynamics, Vol. 2, Nol 1, 1973, 3 10 kiu, S. C. and Neghabat, F., A Cost Optimization Model for Seismic Design of Structures, The Bell System TechnicalJournal, Vol. 51, No. 10, 1972, 2209 2225 Shah, H. C. and Vagliente, V. N., Forecasting the Risk Inherent in Earthquake Resistant Design, Proceedings of the International ConJerence on Microzonation, Vol. 11, 1972, 693 718 Whitman, R. V. and Hein, K. H., Damage Probability for a Water Distribution System, Proceedingsof the ASCE TCLEE Specialty Conference on Lifeline Earthquake Engineering, August 31Y31, 1977, 410-423 Meredith, D. D. et al., Design and Planning oj Engineering Systems, Prentice-Hall Inc., New Jersey, 1973 Himmelblau, D. M., Applied Nonlinear Programming, McGrawHill Inc.. New York, 1972