*Corresponding author. Tel.:#44-161-295-3817; fax:# 44-161-295-5559.
E-mail address:p.a.scarf@salford.ac.uk (P.A. Scarf).
A framework for maintenance and replacement of a network
structured system
Philip A. Scarf
!,
*, Harry H. Martin
"
!Centre for OR and Applied Statistics, University of Salford, Salford M5 4WT, UK
"Department of Technology Management, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, Netherlands Received 1 February 1999; accepted 27 January 2000
Abstract
The asset management problem of a network owner operating in a regulatory climate is discussed in this paper. Typically, the networks of interest are water distribution systems, gas pipeline networks, and electricity supply networks. We look at how network investment projects, relating to refurbishment or replacement of existing assets and network expansion, may be prioritised under capital rationing. The objectives for prioritising such projects will be in#uenced by the regulator; the regulator may set future performance targets, and set caps on charges and capital expenditure. The implications of capital rationing and the relationship between capital investment and future performance and operating costs are considered through the notion of the marginal`costaof delaying projects. The e!ect of the uncertainty in the information that the owner currently possesses about the network is also addressed. An example is given to illustrate the framework presented. ( 2001 Elsevier Science B.V. All rights reserved.
Keywords: Maintenance; Replacement; Networks; Asset management; Capital investment; Regulatory climate
1. Introduction
Network-structured systems, or networks for short, distribute`productsasuch as gas, water, and electricity. These networks typically consist of conduit components (pipes and cables) and control components (pumps, valves, transformers and switchgear), and they link few producers to numerous consumers. The network owners in the UK are retailers of the distributed product, with the network itself forming part of their retail infratructure.
Due to their size and age, the investment in the replacement and maintenance of networks is high. Therefore, the management of a network requires a long-term view of issues such as: the consequences of network failures; progress in network techno-logy; and the development of demand. In many countries, network owners are now privately owned, regulated industries. The e!ect of this is that network owners have to consider their pro" t-ability in a market with charges and supply-relia-bility targets set by the regulator. Therefore, e$cient management of the network and its assets is crucial for pro"tability. The resulting "nancial context for the network owner will encompass in-stallation (refurbishment, replacement and expan-sion) costs; maintenance costs and other
consequences relating to operation of the networks; the regulated income; and the interests of share-holders.
Since there are many network types (di!ering technologies, di!ering products) it is natural to"nd a common approach to network asset management in which the speci"cs of the networks themselves are reduced to a minimum. Thus, we will discuss net-work asset management (NAM) in a general context. Several levels in the NAM decision-making process must be recognized; these we will refer to as decision elements. The decision elements are set out in the conceptual framework described in the next section. Many approaches which consider individual deci-sion elements that have some relation to network asset management in the water industry have been reported [1}6]; for a recent discussion of asset man-agement in the electricity supply industry see Hos-king et al. [7]. However, there has been little work that considers a consistent and complete framework of network asset management. For the most part, this paper concentrates on project planning and release, and looks at how capital investment can be managed to maximise share-holder wealth or net-work performance while meeting certain regulatory and budgetary constraints. Replacement models for project planning and release are described in Section 3. These models are discussed in the context of various decision criteria, and the role of information uncertainty and investment risk is addressed. In Section 4 we present an example to illustrate the framework.
2. The network asset management framework
The ultimate purpose of the network asset man-agement (NAM) framework proposed is to support:
f project planning and release.
The other decision elements in the framework are:
f component identi"cation and evaluation; f maintenance concept design;
f performance measurement.
In this section we introduce and discuss these ele-ments of the framework.
2.1. Component identixcation and evaluation The refurbishment and replacement of existing assets will naturally focus on components of the network. It is expected that a component of the network will be de"ned in terms of thecomponent type, its locationin the network, and its state, and that these will need to be identi"able within a net-work information system (e.g. see Fig. 1). A compon-ent type we de"ne as a physical item whose description is unique when considered outside the context of the network; thus we think of component types as items prior to their setting in the network, e.g. a coil of 100 mm plastic`gas maina. The loca-tion of a component will be characterized by the environmental conditions in which it is installed; and its logical and geographical location. Thestate of a component will be de"ned by its operating intensity, and its current age and/or condition. This information for all components will thus form an evaluation of the`stateaof the network. Compon-ent idCompon-enti"cation will be carried out once only during the life of the component, while component evaluation may be updated on as as-required basis.
For certain networks this element of the frame-work may be trivial }the state of electricity net-works is generally well understood by the network owner otherwise there would be serious concern for safety. For other networks there may be uncertain-ty regarding the state of large parts of the newtork
} UK water supply networks originate, for a large part, from the 19th century and the state of large parts of the network may be unknown, certainly in terms of age and condition of compo-nents and occasionally in terms of component type.
2.2. Maintenance concept design
Fig. 1. Typical rural electricity networks (part) drawn by a network simulator [8]. 500 m squares; (h) infeed; (j) loadpoint; (£) open point; (***) cable; (- - - -) overhead line. (a) 38.9 km of 415 V with 124 load points and 2976 consumers. (b) 481 km of 11 kV with 390 loadpoints and 79950 consumers.
The maintenance concept design will comprise maintenance activities to be carried out on particu-lar components and component groups. The main-tenance activities will be de"ned through the choice
of appropriate maintenance rule for a component failure mode, the level of maintenance intervention, and the extent to which shared execution of indi-vidual maintenance activities is appropriate. Four elementary maintenance rules can be distinguished: failure-based maintenance (maintain on failure only); time-based maintenance (maintain on failure and at"xed times); use-based maintenance (main-tain on failure and when speci"ed levels of com-ponent use are reached); and condition-based maintenance (maintain on the basis of measured condition). In simple terms,"ve levels of mainten-ance intervention can be distinguished: inspection; service (for example, routine adjustment); minimal repair; refurbishment (overhaul); and replacement. The various failure modes of components, their appropriate maintenance rules and the proximity of components will determine the extent to which individual maintenance activities can be grouped. Key components may be subject to condition based maintenance (CBM) in which the operating inten-sity and environment will be important factors.
The interaction between the design of the main-tenance concept and the identi"cation of network components will be signi"cant. For example, for sections of the network on which the network owner chooses to do only minimal repair on failure, there is no need to describe, in the framework, these sections in terms of their components. On the other hand, use based maintenance (UBM) and condition based maintenance (CBM) are implemented for particular components, and therefore, appropriate subdivision of the network into its components is necessary.
2.3. Performance measurement
will not. The performance of the network will impact on costs through the regulation of charges to the customer; the regulator may impose penal-ties for poor performance.
To support project planning and release it will thus be necessary to estimate the performance of the network before and after the completion of a project. Thus by performance measurement we mean calculating, for each potential project, the performance implications of the project for the network as a whole. For example, the network simulator eaNSF developed by Brint et al. [8] does this in the context of electricity supply networks.
2.4. Project planning and release
This decision element aims to schedule network projects (expansion, replacement and refurbish-ment activities) given certain planning constraints. The objectives for this element will have to be clearly de"ned by the network owner. Decisions here may relate to prioritizing of projects, grouping of projects, shifting the moment of execution of projects, budget adjustment (e.g. a network owner may be allowed to alter charges to meet "nancial demands, or to justify savings or an additional budget for large planned projects); overlaying of projects with other network owners (it may be even worthwhile establishing a specialized institutional body to coordinate projects with setups common to other network owners). It will be necessary to
re-#ect the risk of projects in the capital budgeting and this can be done through appropriate choice of discount rates or through portfolio analysis. Marketing analysis relating to future capacity and capability would also in#uence the decisions here.
3. Replacement models for project planning and release
We describe a number of replacement models that can be used in decision making relating to project planning and release. We "rst outline a simple model which will form the basis of the capital rationing models discussed later. This model was introduced by Scarf and Hashem [10] in vehicle#eet replacement. We use the term
replace-ment model throughout because, in the most simple cases, a project will generally correspond to the`replacementaof a component.
3.1. A simple replacement model
Consider a"xed horizon of lengthhover which we will evaluate the consequences of a potential project,P. A potential project may be the replace-ment or refurbishreplace-ment of a component or group of components which comprise part of the network, or a network expansion, or a network re-design. De"nef1
t to be the operating cash#ow (or perfor-mance) relating to P in year tafter the release of projectP. De"ne f0
t to be the baseline operating cash #ow (or performance) relating to P in year
t(that is, relating to that part of the network to be replaced or refurbished underP). For network ex-pansion projectsf0t"0. LetC('0) be the capital cost of projectP.
Suppose the consequences of project release are measured in terms of cash#ows, and let l be the appropriate discount factor. For clarity, we will take income cash#ows as negative and expenditure cash#ows as positive, and we will assume that all cash#ows are incurred at the year end. If project
P is released in year x from now then the total cash#ow overhyears from now will be
x~1 + t/1
f0
tlt#lx
A
h~x+ t/0
f1
tlt#C
B
. (1)If projectPis not released then the total cash#ow over the horizon will be
h + t/1
f0 tlt.
De"ne the`gainafrom releasing projectPin year
xto be the di!erence between these cash#ows:
g
P,x(h)" h + t/1
f0 tlt~1
!
G
x~1+t/1
f0
tlt#lx
A
h~x+t/0
f1
tlt#C
BH
"+h
t/x
Release of projectPin yearx(x(h) will be recom-mended ifg
P,x(h)'0. We can optimize project re-lease by choosing that x for which the gain is a maximum and positive. If g
P,x(h))0 for all
x"1,2,h then release of project P will not be
recommended (over the horizon).
If, alternatively, the consequences of project release are measured in terms of performance, then the gain from releasing project P in year x
will be
g
P,x(h)" h + t/x
(f0
t!f1t~x).
Here we assume that a reduction in the perfor-mance measure, f, represents a performance improvement, which is appropriate when perfor-mance is measured, for example, by the number of interruptions to supply per customer per annum. Note that the performance (or cash#ow) in the year in which the projectPis released,f10, can be used to model reduced performance during execution ofP. Typically, we would expect f0
t'f1t for all t*1
}otherwise projectPwould not be considered for release. Theng
P,x(h)'0 for allx}except perhaps forxclose tohwhenf10is large}andg
P,x(h) will be maximum whenx"1. We may even"nd in prac-tice that f1
t"f1andf0t"f0for all t*1; that is the release of a project implies a one-o! improve-ment in performance from a"xed baseline.
For a large system comprising many potential projects, the outcome of this modelling approach will depend on how the consequences of project release are measured. With consequences in terms of cash#ows, the outcome will be a list of projects that should be released along with their optimum release times, and a list of those that should not. With consequences in terms of performance, the outcome will be a list of projects that should be released immediately, and a list of those that should not. Of course, the release of projects will, in both cases, be limited by the budget for capital expenditure.
The measuresf1
t andf0t will be determined essen-tially through knowledge of the failure processes and the maintenance concept. Such knowledge about the failure processes may be based on objec-tive data, or subjecobjec-tive data or a combination of
both. Methods for handling the subjective data of
`expertsain reliability, and their combination with objective data, are numerous (e.g. see [11]). In addition, the `inconvenience costa that makes a political decision an economic one may be cal-culated as described in Christer and Scarf [12]. The consequences of the adoption of a particular main-tenance concept may also have to be assessed sub-jectively. Income associated with the operation of certain components may be part of cash#ow func-tions, particularly for network expansion projects. Where a project release implies that the condition of some (new) component is monitored, the cost of monitoring and the monitoring pay-back contrib-ute to future cash#ows or performance. Thus the use of monitoring will in#uence decision policy through its consequences.
In the case of investment appraisal for network expansion, we suppose that f0
t"0 for all t. In cash#ow terms, the expansion project would be released in yearxif
!+h
t/x
f1
t~xlt!lxC'0,
provided rational economic policy is operating. For network expansions (and re-design) mainten-ance concepts for components will need to be deter-mined based on design and environment, and the revenue cash#ows (operating expenditure and income) determined. A marketing model must also be available that can generate a demand function and the associated revenues; thenf1
t for expansion can be estimated, with some uncertainty. Note that network expansion could not be considered in per-formance terms in this way (because the gain will always be negative).
3.2. Prioritizing project release
We now consider how the simple model relating to individual projects can be used in planning and release over the horizon (0,h). Letg
g
ij(h)/Ci, whereCi is the capital cost of projecti. If rational decision criteria are to be used to deter-mine policy then projects at the top of the list should be given priority, since they would be asso-ciated with the largest expected gain over the planning horizon. Under capital rationing with a"xed budget, this project priority list would indi-cate which projects can be released in the current year.
For consequences considered in cash#ow terms, appropriate discount rates may be chosen to re#ect the investment risks of projects. A higher required rate of return (smaller discount rate) might be imposed on network expansion projects than on replacement of existing assets. The capital asset pricing model (e.g. [13]) provides a means for determining discount rates in these circumstances. In the case of replacement of existing assets such variation in the choice of discount rates might also be used to re#ect uncertainties regarding new technology.
The capital costs of projects may be formulated to include shared set-ups, and also the cost (conse-quences) of a poor replacement. When an oppor-tunity arises to share a set-up cost, then optimal policy for the network may be re-calculated. This may change the project priority list. Opportunities themselves may arise through: the failure of a component necessitating remedial work; a repair initiated through condition monitoring; the combination of projects on the same network; or combination of projects on di!erent networks. 3.3. A capital rationing model
Suppose the network owner wishes to prioritise project release over the next kyears (k)h). This problem may be approached using linear program-ming (LP), similarly to that proposed in vehicle
#eet management by Karabakal et al. [14]. Sup-pose the capital investment budget for year j is
B
j(j"1,2,h). Introduce the indicator variable
x
ij which takes the value 1 if projectiis released in yearjand 0 otherwise. Then we seek those values of
x
ij (i"1,2,n;j"1,2,k) which maximize the total gain over (0,h) of all projects released subject to the constraints that the capital investment budget is not exceeded in each year. That is
maximize
n + i/1
k + j/1
x
ijgij(h)
subject to
n + i/1
x
ijCi)Bj for allj"1,2,k; (2)
k + j/1
x
ij)1 for alli"1,2,n; (3)
x
ij"0, 1.
Constraint set (2) ensures that the budget for year
jis not exceeded. The annual budgets would re#ect that capital available once imperative projects have been "nanced, perhaps for reasons of health and safety legislation. Constraint set (3) ensures that projectiis released at most once over the planning horizon. Note that if an individual project has negative gain whatever its execution time, then the contribution to the objective function from this project will be greatest when this project is not released over (0,k). Tax considerations in particular contexts will need to be taken into account and modelled; as these are contextual, discussion is omitted. Typically, such planning may be informa-tive over the planning horizon, but only decisions relating to the immediate future (one to two years) would be acted on. Therefore, policy would be continually updated, implying a`rolling horizona
approach. Where a network consists of many iden-tical components, the modelling of project planning may be extended to the case in which a proportion of `similara projects are released in a given year. This could be done by formulating the capital rationing model (CRM) as a mixed programming problem.
3.4. Modelling dependence in replacement costs
While the dependence between the capital costs of di!erent projects may be considered simply using the concept of shared set-up, the CRM model above ignores dependence in operating cash#ows between components. For example, a major expan-sion project, while not replacing existing assets, may have signi"cant operating cost or performance implications for particular assets: the building of a large ring main in a water supply network is one such example.
Essentially, if two projectsP
1andP2interact in this way, then new projects P@
1"(P1, notP2),
P@
2"(notP1,P2), andP@12"(P1,P2) would have to be introduced, along with the constraint to en-sure that at most one ofP@
1,P@2, andP@12is released over the planning horizon. While this approach may lead to a signi"cant increase in the number of
`projectsa in the CRM, in principle the solution procedure would remain unchanged. The existence of future-cost dependencies between projects would have to be identi"ed by the network owner. This may be extremely di$cult in practice. However, such dependency would very much characterize the network replacement problem, and therefore, the approach described is an advance over current methods. A similar approach has been taken by Santhanam and Kyparisis [15] in modelling dependency in the project release of information systems. It is possible that it may be optimal to release both P
1 and P2 during the planning horizon, but not simultaneously. This presents a more di$cult modelling task, without introduc-ing many pseudo-projects, that is. For example, we could consider: releaseP
1at timesandP2at time
t; however, for k"10, say, this would mean the introduction of 25 variables, x
(P1P2)(s,t), for the
P
1,P2 decision alone!
3.5. Cost consideration of sub-optimal policies
For reasons of budgeting constraints or technical delays, the release of some recommended project at some optimal execution moment xH may not be possible. In such cases, it would be informative to have an indication of the extra cost to be incurred in revenue expenditure because of lack of capital
expenditure; this is the marginal increased revenue expenditure due to delayed release (see Ref. [12]).
Given the CRM, and focusing on cash#ows, the operating cost consequences of capital rationing can be determined by calculating the delay asso-ciated with each project as a result of capital rationing. The revenue cost implications due to this delay would from expression (1) be
g
wherex@is the execution time for the project under capital rationing. The marginal increase in revenue expenditure would be found by summing over all projects. In a similar manner, the marginal increase in revenue expenditure due projects delayed in year j, dc
j, could be found by summing (x) over all projects withxH"j. Marginal reductions in capital expenditure under CR, dC
j, could also be cal-culated.
The capital savings and increased revenue expen-diture under CR are in part due to delayed execu-tion and the rest due to non-execuexecu-tion. If the marginal increase in revenue expenditure in yearj, dc
j, increases withjthen this is indicative of the fact that capital investment is too low to control operat-ing costs in the long term. If thedc
j are approxim-ately constant with time then this indicates that capital expenditure is su$cient to maintain the current level of operating cost. In these circumstan-ces, consideration of dC
j will indicate how much more capital investment would be required to reduce revenue expenditure to the optimum level.
When the network operator is interested in maximising performance, the marginal reduced performance due to delayed release can only be considered when comparing alternative budget contraints.
3.6. Choice of decision criteria
assets and network expansions. All criteria might be considered using a MCDA approach (e.g. [16]), or using the "nancial appraisal pro"le approach of Le#ey and Morgan [17]. They may also be considered through suitable constraints in the capital rationing model. A simple approach would consider, for example, two project priority lists, one on the basis of cash#ow, the other on the basis of performance; projects high on both lists would then be candidates for release.
3.7. Modelling the uncertainty in the`costsa
Uncertainty in the cash#ow/performance model parameter estimates, re#ecting the extent of cur-rently available information about particular com-ponents and potential projects, and the extent of technological developments (new materials and techniques), may be propagated through into uncertainty in the gain function,g(h). This would be most easily done using the delta method [18]; see Baker and Scarf [19] for an example of this in maintenance. The variance of the gain, as well as the expected gain, may then be used to produce the project priority list and those projects for which the expected gain is high and the uncertainty in the gain (variance of the gain) is low are candidates for release; these projects would be viewed as sound investments. Markovitz [20] is the classic reference here; for a more recent discussion see Booth and King [21].
Where there is no data regarding a potential project, there will be no objective basis for deter-mining if and where the project lies on the project priority list. One possible approach to this problem would be to use data relating to other projects that are similar in design. Also subjective data may be collected, and used to update component data for the whole network in the manner described in O'Hagan [22] and Goldstein and O'Hagan [23] in the context of sewer networks. These methods are particularly useful for multi-component systems in which there is only limited data for a limited number of individual components. On the other hand, it may be that the income cash#ow may be deterministic in some situations. For example, expansion of the network may be initiated by legislation, and that the compensation for the
investment costs are "xed and predetermined per customer connection.
How uncertainty in the information we have a!ects the segmentation of the network will also require some attention in future work. The e!ect of such structural uncertainty on the outcome of the modelling exercise is a topic of signi"cant current interest in Statistics. A simple approach might con-sider a simple sensitivity analysis; the more system-atic approach of Draper [24], which assigns a prior distribution over the structural model types, lends itself more naturally to a knowledge-based approach within an information system.
4. Example: Electricity distribution networks
The 11 kV overhead networks are a major contributor to performance of the UK electricity supply network. The network owner will consider typical policy alternatives for these networks as follows:
f Upgrading the construction of 11 kV spurs.
f Upgrading the construction of 11 kV main branches.
f Removal of the 33 kV network, that is going straight from 132 to 11 kV.
f Refurbishment of lines, for example: restringing, replacing insulators, replacing poles.
f Using covered conductors rather than bare wire for 11 kV.
f Undergrounding 11 kV lines}cables then tend to be more reliable but have longer repair times. f Repair team deployment }weather is a signi" -cant factor in overhead network performance, as is time taken to reach a fault. Therefore deploy-ment of teams at high risk periods can provide bene"ts.
f Alteration to the protection settings or method of protection.
Following the framework, we would in principle: (1) identify and evaluate the network in
compon-ent terms;
(2) design the maintenance concept;
(3) estimate the performance of the network; (4) identify a potential project;
(5) estimate the performance of the network on completion of the project;
(6) repeat (4) and (5) for other performance im-provement independent projects;
(7) rank projects by their performance improve-ment characteristics (or set-up and solve the CRM);
(8) investigate and report the relationship between annual budget, say, and annual performance improvement.
For electricity supply networks, (1) and (2) above will be well established. The network simulation facility (eaNSF) program [8] has been developed to do (3) and (5), and calculates expected network performance indicators and cash#ows over a num-ber of years for speci"c networks. An implementa-tion of the framework would choose between di!erent policy alternatives, determine which parts of the network to apply them to, and provide deci-sion support in budget setting.
We anticipate a number of di$culties in this approach. Information regarding component relia-bility may be sparse}knowledge about how failure rates alter if, say, a line is refurbished may be di$cult to quantify}and although some data may be available, it may be di$cult to disaggregate factors such as geographical location and weather conditions. There will be uncertainty in the net-work performance indicators; the initial problem is how to relate interruptions to the number of cus-tomer-minutes lost, or to operational costs. This becomes more di$cult the further into the future that predictions are required. It may also be neces-sary to disaggregate the network into regions in order to keep the number of projects within reason-able limits. With a large number of alternative projects, a`what}ifaapproach may be taken. Also, constraints must be introduced into the project release decision-making problem to ensure that projects which are alternatives (for example, under-grounding a 11kV line and refurbishment of the same overhead line) cannot be released`togethera.
Additionally, the likely degradation in the network performance during execution of projects would need to be estimated.
5. Discussion
The companies running the distribution net-works are under constant pressure to reduce operating costs while maintaining or improving standards. Given the maturity of these industries, the management of performance and operating costs rather than carrying out network expansion is the main goal. As the distribution networks typi-cally form a monopoly resource and will remain so for the foreseeable future, regulators limit charges and have introduced penalties for poor network performance. Hence the main objectives of network owners are: maximising average network perfor-mance; minimising expenditure; bounding yearly expenditure; avoiding a disastrous performance, for example, of the type recently experienced by Yorkshire Water, and with power supply in Auckland, New Zealand.
This paper attempts to address these issues through a complete framework for network asset management, which is the "rst step in the development of a model for NAM. The framework is motivated by the need to bridge the gap between technological reasoning and economic decision making; that is between engineering judgement, which provides the basis for maintenance concept design, performance measurement, and the
identi-"cation of potential replacement/refurbishment pro-jects, and the economic decision making of the network owner. The advantage of this framework is that it takes a broad view of the entire process, is not motivated by a particular problem, and attempts to take a top down view of NAM. It is expected that the framework will be useful for di!erent decision-makers both within the network owner at di!erent levels of management, and across di!erent network owners. The framework would also be useful for: the valuation of the assets of the network owner; and for demonstrating that the owner is employing state-of-the-art management of its assets.
ease of implementation within a network informa-tion system. Publicainforma-tions to date in this area, on the other hand, deal with particular aspects of network maintenance and replacement and invariably describe complex models. Thus, an aim of this pa-per is to describe simple models in a complex con-text. The next step is to build a prototype model that looks at a small scale network using the elements of the framework outlined here, initially concentrating on refurbishment and replacement decision-making.
Acknowledgements
The authors would like to thank colleagues at KEMA, Transport and Distribution department, and Tony Christer for his helpful comments on earlier drafts of the paper. The paper has been considerably improved through the helpful comments of the referees. This research has been supported in part by EPSRC grant number GR/L20801.
References
[1] J.H. Stacha, Criteria for pipeline replacement, Journal of the American Water Works Association 70 (1978) 256}258.
[2] T.M. Walski, A. Pelliccia, Economic analysis of water main breaks, Journal of the American Water Works Asso-ciation 74 (1982) 140}147.
[3] A.J. Kettler, I.C. Goulter, An analysis of pipe breakage in urban water distribution networks, Canadian Journal of Civil Engineering 12 (1984) 286}293.
[4] S.A. Andreou, D.H. Marks, R.M. Clark, A new methodo-logy for modelling break failure patterns in deteriorating water distribution systems: Applications, Advanced Water Resources 10 (1987) 11}20.
[5] D. Li, Y.Y. Haimes, Optimal maintenance-related decision making for deteriorating water distribution systems: 1, Semi-Markovian model for a water main, Water Re-sources Research 28 (1992) 1053}1061.
[6] D. Li, Y.Y. Haimes, Optimal maintenance-related decision making for deteriorating water distribution systems: 2, Multi-level decomposition approach, Water Resources Research 28 (1992) 1063}1070.
[7] R.P. Hoskins, A.T. Brint, G. Strbac, A structured approach to asset management in the electricity industry, Utilities Policy 7 (2000) 221}232.
[8] A.T. Brint, W.R. Hodgkins, D.M. Rigler, S.A. Smith, Evaluating strategies for reliable distribution, IEEE Computer Applications in Power 11 (1998) 43}47. [9] C.W. Gits, Design of Maintenance concepts, International
Journal of Production Economics 24 (1992) 217}226. [10] P.A. Scarf, H. Hashem, On the application of an economic
life model with a "xed planning horizon, International Transactions in Operational Research 4 (1997) 139}150. [11] J.M. Van Noortwijk, R. Dekker, R.M. Cooke, T.A.
Maz-zuchi, Expert judgement in maintenance optimization, IEEE Transactions in Reliability 41 (1992) 427}432. [12] A.H. Christer, P.A. Scarf, A robust replacement model
with applications to medical equipment, Journal of the Operational Research Society 45 (1994) 261}275. [13] M.C. Ehrhardt, The Search for Value: Measuring the
Company's Cost of Capital, Harvard Business School Press, Boston, 1994.
[14] N. Karabakal, J.R. Lohmann, J.C. Bean, Parallel replace-ment under capital rationing constraints, Managereplace-ment Science 40 (1994) 305}319.
[15] R. Santhanam, G.J. Kyparisis, A decision model for inter-dependent information system project selection, European Journal of Operational Research 89 (1996) 380}399. [16] R. Santhanam, G.J. Kyparisis, A multiple decision criteria
models for information system project selection, Com-puters in Operations Research 22 (1995) 807}818. [17] F. Le#ey, M. Morgan, A new pragmatic approach to
capital investment applications: The "nancial appraisal pro"le model, International Journal of Production Econ-omics 55 (1996) 321}341.
[18] C.R. Rao, Linear Statistical Inference and Its Applications, Wiley, New York, 1973.
[19] R.D. Baker, P.A. Scarf, Can models to small data samples lead to maintenance policies with near-optimal cost? IMA Journal of Mathematics Applied in Business and Industry 6 (1995) 3}12.
[20] H.M. Markovitz, Portfolio selection, Journal of Finance 7 (1952) 77}91.
[21] P. Booth, P. King, The relationship between"nance and actuarial science, in: D.J. Hand, S.D. Jacka (Eds.), Statistics in Finance, Arnold, London, 1998, pp. 7}40.
[22] A. O'Hagan, Robust modelling for asset management, Journal of Statistical Planning and Inference 40 (1994) 245}259.
[23] M. Goldstein, A. O'Hagan, Bayes linear su$ciency and systems of expert posterior assessments, Journal of the Royal Statistical Society Series B 58 (1996) 301}316. [24] D. Draper, Assessment and propagation of model
![Fig. 1. Typical rural electricity networks (part) drawn by anetwork simulator [8]. 500 m squares; (of 415 V with 124 load points and 2976 consumers](https://thumb-ap.123doks.com/thumbv2/123dok/3115098.1378394/3.544.49.262.85.523/typical-electricity-networks-anetwork-simulator-squares-points-consumers.webp)
