risk analysis and simulation 0066v1-2

8/17/2019 Risk Analysis and Simulation 0066v1-2

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Contents

1. Analysing the risks in construction projects

2. SERC recommendations 2.1 Contract strategy2.2 Risk assessment2.3 Risk allowance2.4 Risk management

3. Risk allocation

4. Bearing the risk

5. Methods of dealing with risk

5.1 The qualitative risk analysis stage5.2 The quantitative risk analysis stage5.3 Sensitivity and probability analyses5.4 Decision trees

6. Simulation 6.1 Monte Carlo simulation6.2 Probabilty Density Distribution (PDD)

This paper was written by Roger Waterhouse MSc FCIOB MIMgt MAPM MSIB

© The College of Estate Management 2004

Paper 0066V1-2

Risk analysis and simulation


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Risk analysis and simulation Paper 0066 Page 3

1 Analysing the risks in construction projects

The elements of risk and uncertainty are present throughout all stages of aconstruction development.

At the inception stage, a client has to decide: is a new building really necessary? Orwill expansion or refurbishment of the present premises suffice? What are the

drawbacks of moving to a new location? How much should be invested and how longwill it take to reap the benefits? All these factors will need to be carefully consideredand their associated risk elements analysed. The probability of each risk eventcoupled with its potential consequence should receive special attention.

The subsequent stages of feasibility, strategy, pre-construction, construction andcommissioning will each bring with them a variety of risks and uncertainties. It willbe left to the project manager to evaluate and manage these risks.

The ‘what if’ scenarios will each need analysing, the risks identified and the variablescalculated as far as possible. Thus the unpleasant surprises which accompany allconstruction projects should be kept to a minimum.

The project manager will need to appreciate that the process of risk identificationinvolves not only recording that a risk exists, but also assessing the consequences ofthat event occurring. The assessment process therefore commences with the riskidentification and then each risk must be classified in terms of possible consequence.A single risk may only have minor consequence, but a combination of seemingly‘minor’ risks could produce a major consequence on the project. Hence theimportance of a comprehensive analysis.

In order to manage the risk, the project manager will need not only to identify the riskevents but to consider how to prevent, avoid, contain or transfer the respective events.

Collectively these constitute risk management. (See Figure 1.)

Those with the highest element of risk or uncertainty will probably require the closestscrutiny, but this will depend very much on the criticality of the event itself.

It is therefore important to distinguish the sources of risk from their effects.Irrespective of their sources all risks may affect the project objectives and the keyelements of time, cost, and quality.

Risks may be characterised by the following factors:

Risk event – what may happen to the detriment of the project.

Risk probability – the likelihood the event may occur.

Amount at stake – the amount of the possible loss.


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Some argue that there is a difference between risk and uncertainty; others howevermaintain that the difference is unimportant. Those who hold the former viewpoint

believe that risk involves an assessment based upon historical data or experience. Adecision is made based upon the probability of a particular event occurring; in otherwords, a forecast is made using past data and therefore within some degree of

certainty. On the other hand uncertainty is when no historical data or experienceexists.

FIGURE 1 The risk management process


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2 SERC recommendations

According to a SERC report, ‘Risk Management in EngineeringConstruction’ (1986), some of the main recommendations for dealing with risks areas follows:

2.1 Contract strategy

Attention to contract strategy based upon systematic consideration of risk canachieve significant cost savings for a project. The proposals for funding aproject should therefore include recommendations on contract strategy.

All too often risk is either ignored or dealt with in an arbitrary way: simplyadding a 5 percent ‘contingency’ on to the estimated cost of a project istypical.

2.2 Risk assessment

The greatest uncertainty is during the early stages of a project, which is alsowhen decisions of greatest impact are made. Risk must be assessed andallowed for at this stage.

The client’s departments and advisers should operate as a single team to avoidthe institutional risk of incomplete commitment and inconsistent decisions.

Flexibility in project design and the risk of later changes should be consideredin detail before completing proposals for sanctioning.

The quantitative techniques can be used to analyse probabilities and thesensitivity of predictions to uncertainties in estimates to give a much moreaccurate assessment of risks.

2.3 Risk allowance

Risk techniques are widely used in other industries. The techniques are nowwell within the reach of small companies, requiring only a microcomputer tobe put into action.

The analysis should be carried out by those trained to do so jointly with projectplanners and cost estimators.

On many construction projects, the client would be deceiving himself if heused just one single figure estimate of cost and time for appraisal and fundingdecisions. Ranges of estimates should be used, including specificcontingencies and tolerances for uncertainty.

Delay in completion can be the greatest cause of extra cost, inconvenience andof loss of financial return. The first estimate of cost benefits should be basedon a realistic programme for a project, so that the potential effects of delayscan be predicted realistically.


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2.4 Risk management

For high-risk contracts, project sponsors should specify the allocation of riskwhen inviting bids and require tenderers to state their provision for risk (byway of available resources) in their bids. Project sponsors should also considerselecting the contractor on the basis of ‘minimum acceptable risk’ rather thanlowest price. Risk analysis allows such a criterion to be used.

Clients and all parties involved in construction projects and contracts benefitgreatly from reduction in uncertainty prior to their financial commitment.Money spent early buys more than money spent late. Willingness to invest inanticipating risk is a test of a client’s wish for a successful project.

Risks change during most projects. Risk management should therefore be acontinuing activity throughout the life of a project.

Much can be learned about the implications and management of project riskwithout extensive numerical analysis. Risk analysis is essentially a brain-

storming process of compiling realistic forecasts and answers to ‘what happensif?’ questions.

Competitive tendering coupled with traditional contractual arrangements limitthe realistic management of risk. The pressure is always on those bidding forcontracts to keep their tender prices as low as possible, which can put boththem and their clients at great financial risk if things go wrong. When someprovision has been made for eventualities, it is often buried in the total bid.This hinders the effective management of risk and militates against asystematic and equitable basis of payments.

In conclusion, risk identification involves indicating all potential risks the projectmight face and assessing their impact and probability of occurrence to decide whichrisks need to be managed.


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FIGURE 2 Risk identification – cause and effect

Possiblecause

No.Event

Possibleeffect

Lack of

compound

Disruptionduring

investigations

Absence ofnight-

watchman

Delay toproject

Store leftunlocked

1Theft ofmaterials

Increased cost tocontract

Valuable items Rise in insurance

premiums

Poor procedures

Reduction inmoraleon site

Small size Labour andplant idle

Reduction in

workload

Lack ofmaintenance

Concrete sets in

pipe

Lack of cleaning Delay to

programme

Concrete toostiff

2Concrete

pump failure

Increase in costto contractor

Aggregate toohigh

Need to replacepump

Lift too high

Removeconcrete

already placed

Additional stopend

required in pour


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3 Risk allocation

It is clear, therefore, that risks and their effects should be considered throughout theproject by all the parties involved in the project.

In general, the risk should be allocated to these parties on a solid basis ofresponsibility and control in a manner likely to optimise project performance. The

burden of responsibility for the allocation of risks should rest with the projectmanager as the client’s representative – although this will depend to a certain extentupon his delegated authority.

Classification of sources of risk may be made under the following headings: design,construction, environmental, financial, legal, operational, political, and physical.

Ideally, the allocation of risk should be done by the client through the contractdocuments. However, this does mean that a full analysis of all identifiable risksshould be executed at the very start of a contract. Some might argue this is notaltogether feasible with modern fast-track projects; nevertheless, the principleremains true that as much of the analysis as possible should be carried out as early aspossible.

The allocation of risk will depend on the type and conditions of contract.Traditionally, it is the main contractor to whom the largest part of the risk isallocated. However, perhaps the greater concern is whether the client and/orconsultants have ascribed responsibilities to the contractor which are not properlywithin his scope or control. If so, it will invariably result in tension between theparties.

During this stage all risk is assessed in terms of its probability and magnitude or size.Quite often the number of risks that have a high probability of occurrence and a high

impact is not great in construction projects. However, sometimes there is a long list ofthe high-probability, low-impact risks to be considered and this can make the riskanalysis difficult. On the other hand, sometimes the number of risks which make amajority of the total risk to be considered is not great. For example, consideration ofseven large risks might cover as much as 85 percent of the total. This makes the riskanalysis easier and enables the contractor or analyst to concentrate his attention onthese relatively few critical sources of risk.

The allocation of risks should be based on a thorough appraisal of the relationshipbetween the respective party and risk. Incentives and risk go together. A party thatcarries a risk has the incentive to minimise its impact. The basis for allocation of risksshould be:

control of the risks or their effects by the parties concerned;

ability to perform a task related to the project; or

inability of all parties except the client to accept a certain risk.

Whatever principle governs the allocation of risks between contractual parties, thereis always a danger that this allocation has not been done properly, and therefore a riskwhose allocation is not clear can occur and cause disputes. The probability of disputebetween the parties reduces proportionally with the reduction in the number ofunallocated risks.


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The traditional method for allocating risks in the industry of construction and realestate is as follows:

From client to designer and contractor(s).

From either the client, designer or contractor (main or sub) to insurer.

From client, funder, developer, purchaser or tenant to professionals (architect,

engineer) or contractor by collateral warranty. From main contractor to subcontractor (domestic and nominated).

From contractor (main or sub) to guarantors or sureties.

It is often possible to reduce risks, and this should be attempted before the allocationis made. On most projects, certain risks can be reduced at relatively little cost. Theright time to consider possibilities of risk avoidance or reduction is the early stage ofa project. For example, the risk associated with uncertain ground conditions is alwayshigh. It can be reduced by drilling a large number of exploratory boreholes.

It is also possible to transfer the risk by contracting to another party, whether by

insurance or by its inclusion within the implementation contract (see Figure 4).

FIGURE 3 Risk categories

R I S K S

INVESTMENT DEVELOPMENT POLITICAL DISASTER

Financial Funding National & EU

Funding collapseProject abandonmentDelay to projectCost increase

Interest rate changes

see ‘Investment’ TaxationLegislationEnvironmentalGrants

PlanningGovernmental stability

ExplosionEarthquakeFloodRain

LightningSnow/iceSubsidenceFireHurricane

Commercial Design International

Viability ofdevelopmentMarket changesNational economy

BuildabilityErrorsChangesDesign choiceStability

Joint venturesGovernmentalConsortiumForeign aidLabour movementControlsGovernmental stability

Construction

ProgrammePlanningDesign changes

Site investigationMaterial shortagesSafetyCost increasePlant breakdownNew techniquesResource shortagesStatutory authoritiesPenaltiesTown planningWeather

Operational

MaintenanceEnvironmentalSafetyEnergyLocation


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4 Bearing the risk

Risk should be equated to the expertise, role and reward of the party to whom it isascribed. A developer who bears the risk of success or failure of a project gains theprospect of substantial loss or profit. The consultant whom the developer employswill receive a prescribed fee – no more, no less – for providing specialist advice andservices. The responsibilities he undertakes should be appropriate to that role;

similarly for the contractor and all others in a project team.

When ascribing risk therefore, it is reasonable for a contracting party to bear risk inany one of the following cases:

If the party can cover a risk by insurance, and it is reasonable for the risk to bedealt with in this way.

If the risk is of loss due to the party’s own misconduct or lack of reasonablecare.

If it is in the interests of efficiency to place the risk on the party.

If the economic benefit of running the risk accrues to the party.

An important consideration when allocating project risks is the willingness of therespective parties to take on the risks. The allocation of the project risks,contractually, is generally in the hands of the client.

If the client is unwilling to bear a particular source of risk, he can pass this on to oneor more of the other parties involved in the project, including the managementcontractor in a management contract if he so wishes. Naturally, the client will pay aprice for passing on this risk, although he does not always fully appreciate thepremium he has to pay for this.

The client may also incur a further, additional cost via an impact on the projectobjectives of professionals’ and contractors’ behaviour over the life of the project.For example, quality may suffer, delays occur, or claims may arise that increaseproblems and potentially add to the project’s cost.

Consultants or contractors who are obliged to bear the project risks have thefollowing options:

To continue to bear the risk and manage it for profit, but accept liabilities.

To pass the risk on to a third party.

If vulnerable, to try to recover costs from other parties, including the client.

If vulnerable, to meet liabilities reluctantly or leave the contract or declarebankruptcy.


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5 Methods of dealing with risk

There are several different ways of analysing and dealing with risk. Some of these aredemonstrated diagrammatically in Figures 2, 3, 4 and 5.

The degree of risk may also be affected by the type and form of contract. TheInstitution of Civil Engineers’ ‘New Engineering Contract’ attempts, for example, toadvance earlier models in relation to the analysis and allocation of risk. One of itsaims is to reduce the extent of disputes by all interested parties. Such disputes oftenarise from unclear procedures in the written contract.

The ‘New Engineering Contract’ aims to identify more clearly the risks and theresponsibilities for managing them. The intention is to place the overall managementof risk much more squarely on the shoulders of the project manager (and engineers)and to significantly reduce the role of the lawyers and insurers on constructionprojects. It deals with risk in the following manner:

By using language which is simpler and more understandable than inalternative forms of contract.

By outlining a single procedure for compensating the contractor when a riskoccurs.

By identifying in one list, a standard risk allocation between client andcontractor as well as allowing a tailored allocation of special risks.

By directing the user to select his preferred contract strategy, which has theeffect of drawing attention to the differences in the allocation of risk between

the respective strategies.

FIGURE 4 Classification of risk events

Probability

Consequence

Minor(£0–1000)

Moderate(£1000–10,000)

Major(£10,000–100,000)

Critical(over

£100,000)

MostImprobable

Accept Insure Insure Insure

Improbable(rare)

Accept Insure Insure Insure

Possible Accept Insure InsureChangeMethod/Design

ProbablePartialInsure

Insure InsureChangeMethod/Design

MostProbable

(common)

PartialInsure

Insure InsureChangeMethod/Design


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However, whichever contract is selected, a procedure will need to be adopted fordealing with the particular risks in question. Risk analysis requires a systematicapproach which is designed to suit the model and circumstances in which they areused. The following method is a fairly general one and consists of two distinct risk

analysis stages: qualitative risk analysis and quantitative risk analysis.

5.1 The qualitative risk analysis stageThe two aims associated with this process are:

To identify the risk

To make an initial risk assessment.

The process involves compiling a list of the main risk sources with a description oftheir likely consequences:

Examine list of risk compiled from previous projects.

Investigate potential risks with key project participants.

Determine possible solutions via brainstorming sessions with the project team.

This process not only helps to examine the potential problem areas with a project, butit also brings considerable benefits in terms of understanding the project. It helps tofocus the respective minds of the project team members by provoking thought aboutthe management responses to the risks. A good understanding between the teammembers is also a useful side effect.

Assessments of cost and time improve as the project proceeds, but it should always beappreciated that the most significant decisions are made during the early stages of aproject. Therefore a realistic estimate of the final cost and project duration is required

as early in the project as possible. It is at this stage that all the potential uncertaintiesand risks likely to affect the project should be identified and hopefully fully assessed.

This will also encourage the project manager and his project team to concentrate onstrategies for controlling the risks as well as determining the allocation of risk to therespective parties. Furthermore, it will help identify what additional design andresources are most likely needed.

5.2 The quantitative risk analysis stage

Quantitative analysis often involves more sophisticated analysis techniques whichusually require computer application. These are less subjective, although in manycases the raw data has had considerable subjective input. This method of analysisrequires:

Probabilistic combination of individual uncertainties.

Estimates of uncertainty in predicting the cost and duration of activities.

Computer-generated models and analytical techniques can be useful indicators oftrends and problems for attention; they should not be relied on as the sole guide todecisions. Their accuracy depends on the realism of assumptions made, the skill ofthe model builder and the accuracy of the data used.


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One method for considering project risks is to analyse any risk independently ofothers, with no attempt to estimate the probability of occurrence of that risk. Theestimated effects of each risk can then be accumulated to provide maximum andminimum project outcome values. In other words neither a subjective nor ananalytical value is given regarding the probability of occurrence of the risk event.Instead, each risk event is compounded to determine the possible effect upon theproject and then, by applying a range of maximum (critical) to minimum (minor – seeFigure 4) project outcomes (consequences), the full extent of the particular risk can beseen.

FIGURE 5 The risk analysis process


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5.3 Sensitivity and probability analyses

The next stage, although more complex, is to apply probabilities to the risks andconsider the interdependencies between the risks. Two useful techniques for doing so

are sensitivity analysis and probability analysis. When undertaking either of these,the following should be considered:

Type and size of project

Information available

Cost of the analysis and the time available

Expertise of the analysts.

Sensitivity analysis

This is a technique used to consider the effect upon the project of changes to thoseevents which are deemed a potential risk to the project. A sensitivity analysis requirescalculation of the effects on the project for a range of values of the event changes.The effect on the project (project outcome) is usually expressed in terms of NPV,IRR, time or final cost.

For example, one of the risks on a project is the cost of steel and there is a risk thatthis could increase by 2½, 5, 10 or 15 percent. In this case the project outcome isevaluated for each of these potential cost changes and the result can then be plottedon a graph to show the percentage variation (Risk v Cost change).

The results of a sensitivity analysis can be shown graphically on a ‘spiderdiagram’ (see Figure 6). This example is based upon analysing the possible costs tothe contractor of a contract to construct a motorway. The diagram shows the results ofcalculating the sensitivity of the cost to changes in each risk which could affectproductivity on site. For instance, it indicates the effect upon the overall project costof a decrease in drain-laying output (increase in duration variable, raises costs).

A sensitivity analysis is very useful because often the effect of a small change in onevariable (a cost or a duration, for example) produces a marked difference in theproject outcome. When several risks are being assessed in this way, a ‘spiderdiagram’ provides an effective way of demonstrating risks which are most critical andsensitive. These are the ones the project manager must act upon.

Such an analysis can be performed for all the risks and uncertainties which may affecta project in order to identify those which have a large impact on the cost, time orwhatever the objectives are. This procedure may be used to identify the variables tobe considered for carrying out a probability analysis – see below.

One problem with a sensitivity analysis, however, is that each risk is consideredindependently with no attempt made to quantify their probabilities of occurrence.This procedure is also limited because in reality a variable would not change withoutother project factors changing and this is not shown in the sensitivity analysis.Eventually, when the user has gained sufficient practice, the number of risks in needof consideration can be reduced because those which have a large impact on theproject tend to become more easily identifiable.


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Probability analysis

This is a technique which can extend beyond the limitations of a sensitivity analysisby specifying a probability distribution for each risk and then assessing the effects onthe risk events in total. However, careful interpretation of the results is essential. Oneimportant stage in this type of risk analysis is assessing the range of probabilitieswhich could result.

The use of ‘Monte Carlo’ simulation (random sampling) can be made where

calculation of data inserted into an equation would be difficult or impossible. Thisprocedure may be used in a probability analysis as follows:

1. The variations to the risks being considered are assessed and a suitableprobability distribution of each risk is selected; then –

2. For each risk a value within its specified range is selected. The value should berandomly chosen and be within the estimated probability distribution.

3. To establish the outcome for the project, a calculation is made based on thecombined values for each risk.

4. The process in (3) is repeated several times in order to produce the probability

distribution of the project outcome.

FIGURE 6 Risk v cost change


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More about simulation is described below.

5.4 Decision trees

On all major projects there are several different routes available in order to achievethe objectives. Therefore the client or project manager is faced with a variety ofalternatives.

Decision trees are a useful means to assess the likely impact of uncertainty onmanagerial decision making. They provide a graphic display of the range of possibleoutcomes and can be used to assist making, or else provide justification for, aparticular choice of action(s).

EXAMPLEManagement of a manufacturing industry need to decide whether it is better to build andset up initially either a large or a smaller warehouse. Best available estimates suggest theformer would cost approximately £7m, whilst the latter £5m. Profitability over the firstfive years depends on demand, which although unknown has been researched to eitherincrease from current levels or to remain about the same. An option that is being seriouslyconsidered is to construct the smaller warehouse first and then to expand to the capacity ofthe larger warehouse (at project additional cost £4m) if the market has shown signs ofincreasing after two years. Further objective market research has revealed that theprobability of experiencing increasing demand within the first two years is around 0.25,and the conditional probability of continued rising demand beyond this is 0.5. Theseprobabilities, and associated expected profits, ignoring building costs, after five years ofoperation are displayed in Figure 7.

Is the two-stage development strategy a sensible one? Or is it expected to be moreprofitable to build a larger warehouse immediately, given the likely outcomes after fiveyears?

Note that, by convention, a square node depicts a choice to be made, whilst circularnodes correspond to chance events.

To shed light on the practical problem posed, one method is to evaluate directly theexpected value of the profit involved at each node. This criterion says that theexpected value, or worth, of a profit of £x that materialises with probability p is £(px).Thus, the expected value at each circular node is the sum of all the £(px) quantities.Note that summing just the probabilities p together out of any one chance node has toequal 1, since precisely one branch must occur. (Recall the probability of a certainevent is, by definition, 1.) To find expected values at each square node, it is necessaryto subtract the costs associated with the particular branch. It is straightforward, byworking backwards exhaustively through the entire decision tree to find the mostappropriate decision at each node simply by choosing the branch that maximisesexpected profit at every step. This enables the best overall strategy to beimplemented, or at least to give a more solid basis on which to make decisions thanpurely by intuition alone!


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EXAMPLE (continued) Figure 8 shows the expected values inserted into the decision tree. The uppermost circularnode has expected value £8250k, derived from £12m*0.25 + £7m*0.75. The secondmostchronological decision node (should warehouse be expanded later or not?) has expectedvalue of £8000k, being the larger of £8m*0.5 + £6m*0.5 = £7m, if not expanding, and£14m*0.5 + £10m*0.5 + (–£4m) = £8m, if choosing to expand.

Continuing, the first chronological decision can be seen to be between expected profits of£1250k for building the large warehouse immediately versus £2250k for the morecautious ‘wait and see’ option.

Hence, application of the decision tree method strongly supports building the smaller

warehouse for now in this example.

FIGURE 7 Decision tree: five-year profits, building costs and probabilities

FIGURE 8 Example continued with expected values (in £m) inserted at nodes


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There are some obvious limitations to this technique. Supremely, used naively, onehas to put faith in the accuracy of all the probability figures and cost estimates. If anyof these happen to be seriously wrong, so too might be the recommended decision(s).So in practice how is this surmounted?

The answer is to conduct a sensitivity analysis, similar to that described in riskanalysis above, especially if there are grounds for reasonable uncertainty in the

figures given in the decision tree. Such grounds may be because not all values arebased on well-researched, historic data. Clearly, if costs and probabilities are no morethan wild guesses, the method, used simplistically, has little more to offer than blindintuition. Nevertheless, the method comes into its own when large and complexdecisions are broken into a series of smaller ones, any of which can be subjected tominor changes in associated numeric values to assess its overall impact. For instance,in the above example, one could consider best and worst case scenarios and see whatdifference may be made to the initial decision about size of warehouse.

If fortuitous, it may turn out to be the case that the identical initial decision isrecommended regardless of quite major changes to probabilities and/or expectedprofits further down the tree. Even if not, one can ask questions like if all but the

(say) £14m figure were unchanged from Figure 7, how large or small would it need tobe in order to maintain the overall best choice of building a smaller warehouse tobegin with? Equally, one might pose the question in terms of how likely it must be,instead of the presumed 0.25 probability, for demand to be increasing sufficient to

justify building initially the larger warehouse (again, all other things – besides the0.75 complementary probability – being kept equal).

Another important consideration in the use of this technique is the appropriateness orotherwise of the expected value criterion. As a rule of thumb, the more complex thetree, the better this criterion becomes. A typical structure of a complex tree isillustrated in Figure 9. The reason why this rule holds is because, with small treesinvolving just a handful of decision nodes and probabilistic outcomes, there is not

really scope for expected value to have direct meaning. Again, referring to the aboveexample, one could say that building the large warehouse now gives rise to either a£5m net profit or breaking even (these being, respectively, £12m – £7m, and£7m – £7m). Having simplified the state of affairs to just, in effect, high and lowdemand, neither outcome would yield the implied £1250k. It may be helpful to recallanalogously that one difference between a sample mean and a median is that theformer may not be an attainable value (eg who has 2.4 children?) whereas, in general,the latter measure will be.

As a further use of decision trees, one can employ them to make judgementsinvolving other criteria besides expected value. Specifically, it may be in the bestinterests to use a criterion that minimises the worst loss, or perhaps more

speculatively, maximises the highest profit. Such criteria are called minimin,maximin or minimax, etc and there are circumstances when it is more prudent toapply such a criterion in place of expected value. However, as indicated, for morecomplex decision trees, it is usually sensible to use expected value, as in the longerrun, overestimates and underestimates of expected values tend to balance out,assuming importantly there is no systematic bias in the allocation of costs andprobabilities.

In summary, decision trees can provide a powerful technique for convincing oneself,and others, about the most suitable course of action when faced with uncertainty.

This graphical representation sometimes makes the solution obvious and, by the

addition of estimated costs, values of outcomes and probabilities, provides a basis foranalysing complex problems. Hence this system can help clarify and communicatethe option available to the project manager.


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It can be used in choosing the appropriate procurement route, or the method ofconstruction or even whether to proceed with a project claim. Its more obvious use isby clients in choosing between alternative development projects.

While it is not without limitations (in common with most tools in ManagementScience) a key advantage of this approach to risk and decision is that it encouragesthe client/project manager/decision-maker to assess some degree of probability on an

outcome occurring and to take rational and logical action at each decision point.

FIGURE 9 Typical ‘decision tree’ structure for comlex problem


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6 Simulation

6.1 Monte Carlo simulation

Building and analysing a simulation model will be more effective if a substantialbank of data exists or can be generated. Monte Carlo simulation makes use of randomnumbers to generate random data based on known facts or observations.

Tables of random numbers, as typically found in any statistics textbook, are producedby computer using various algorithms which ensure that all numbers from 00 to 99(usually) have equal probability of occurring at any point on the tables. When usingrandom number tables it is important not to introduce bias: the numbers should beused strictly in order as shown in the examples.

Throughout this section, just a single iteration of the simulation process is described.In practice, a computer would perform literally hundreds or thousands of suchiterations, and produce a summary of all the results. This summary is quite likely toreflect reality, provided of course initial assumptions in the simulation model arevalid!

6.2 Probability Density Distribution (PDD)

Simulating a problem using random numbers

Any simulation model should have a factual basis from which probabilities can bedetermined. Random numbers used for the generation of data are allocated inaccordance with the probabilities.

EXAMPLE 1

Consider a stock control situation where the demand for a particular product varies fromday to day. Observations show that the demand per day recorded over a period of 100

days is as follows:

The probability of a particular demand can be calculated:

Demand per day (items) Number of days

0 2

1 8

2 22

3 34

4 18

5 9

6 7

Demand per day Number of days Probability

0 2 .02

1 8 .08

2 22 .22

3 34 .34

4 18 .18

5 9 .09

6 7 .07


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Probability density distribution table

Random numbers are allocated in accordance with the probabilities. For eachprobability point, one random number is allocated. Thus since the probability of ademand of zero items per day is 0.02 then two random numbers must be allocated,namely 00–01. The next eight numbers are allocated to a demand of one item per day02–09, and so on.

The allocation of random numbers may be simplified by considering the cumulativeprobabilities as follows:

Note the numerical relationship between the cumulative probabilities and theallocated random numbers.

Using the probability density distribution

Data which simulates the demand per day is generated by reading random numbersfrom the tables and looking up the relative demand in the PDD. For example:

In order to simulate the stock control problem, a second PDD is required for the

supply or delivery of goods. The recorded delivery times following placing an order,over 50 deliveries, are as follows:

Demand per day Probability Cumulative probability Random numbers

0 .02 .02 00–01

1 .08 .10 02–09

2 .22 .32 10–31

3 .34 .66 32–65

4 .18 .84 66–83

5 .09 .93 84–92

6 .07 1.00 93–99

Random number Simulated demand per day

84 528 2

64 3

49 3

06 1

75 4

09 1

73 4

Time to delivery (days) Frequency

2 13

3 35

4 17

5 5

Total 70


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The PDD for delivery and allocation of random numbers is as shown:

Simulating the stock control problem

Using the demand and delivery PDDs shown above, run a Monte Carlo simulationover a period of 15 days for a stock control situation where the buffer stocks (level atwhich an order is placed) are 10, initial stock is 15, and the re-order quantity is 20.

Average stock = 11559 = 10.6 units

We could, of course, add in the various costs associated with the ordering and storageof stock, and by varying buffer stock level, re-order quantity etc a number ofsimulations could be carried out to optimise these values.

Queuing models can also be investigated using simulation and decisions regarding thenumber of service points to be installed can be arrived at economically.

Time to delivery(days)

Probability

Cumulative probability

Randomnumbers

2 .19 (13/70) .19 00–18

3 .50 .69 19–68

4 .24 .93 69–925 .07 1.00 93–99

Daynumber

Demandrandom number

Demand Stocklevel

Deliveryrandom number

Days Daynumber

0 — — 15

1 84 5 10 28 3 4

2 64 3 7

3 49 3 4

4 06 1 23

5 75 4 19

6 09 1 18

7 73 4 14

8 49 3 11

9 64 4 7 93 5 14

10 39 3 4

11 89 5 out of stock

12 — — out of stock

13 — — out of stock

14 77 4 16

15 86 5 11

159


23/26


EXAMPLE 2Aggregates are delivered by rail to a large civil engineering contractor. The trains arebelieved to arrive according to a well-known standard probability distribution called‘Poisson’, with a mean of 1.25 per day. Two unloading services are available for amaximum of seven hours per day each and the unloading frequencies applicable to bothare shown below:

a. Use Monte Carlo simulation to predict the arrivals pattern and utilisation of theunloading facilities over a 10-day period.

b. Is a further unloading point justified?

Using a standard formula (that is beyond the scope of this course) for the arrivalprobabilities, the following PDDs can be drawn up:

Arrivals PDD

* rounded to 2 decimal places

Unloading PDD

(Continued)

Unloading time per train (hours) Frequency5 15

6 23

7 36

8 41

9 25

Arrivals per day Cumulative Poisson probability*

Randomnumber allocation

0 0.29 00–28

1 0.64 29–63

2 0.87 64–86

3 0.96 87–954 0.99 96–98

5 1.00 99

Unloading time Frequency probability

Cumulativeallocation

Random number

5 15 0.11 00–10

6 23 0.27 11–267 36 0.53 27–52

8 41 0.82 53–81

9 25 1.00 82–99

140


24/26


EXAMPLE 2 (continued)

Simulation, using a particular sequence of random numbers gives:

which is the time in use divided by time available for use.

A further unloading point is not justified. Indeed the evidence suggests that one unloadingpoint would be adequate.

Daynumber

Randomnumber

Arrivals per day

Randomnumber

Unloadingtime

Utilisation ofunloading facilities

No 1 No 2

(hours) (hours) (hours)

1 56 1 77 8 7 –

2 39 1 73 8 1 7

3 69 2 96 9 7 1

89 9 2 7

4 11 0 – – – 2

5 80 2 49 7 7

36 7 – 7

6 11 0 – – – –

7 19 0 – – – –

8 03 0 – – – –

9 51 1 77 8 7 –10 01 0 – – 1 –

Totals 32 24

Utilisation factor =(32 + 24) × 100

10 × 7 × 2 = 40%


25/26


EXAMPLE 3Question: A digital simulation model is required for forecasting the likely financial performance of acompany offering a specialist technical service to the building industry. The available dataare given in Tables 1 and 2 below.

a. Simulate a set of 12 monthly sales income values. Use the random number

sequence 3, 47, 43, 73, 86, 36, 96, 47, 36, 61, 46, 98 for this purpose.

b. Simulate a set of 12 monthly cost values. The random number sequence 16, 22, 77,94, 39, 49, 54, 43, 54, 82, 17, 37 is to be used.

c. Define the remaining steps in the sequence of calculations which must beperformed in order to complete the simulation.

d. Comment on the apparent situation of the company and make suggestions forimprovement.

DATA

Answer:

PDD Gross Monthly income

(Continued)

Table 1 Table 2

Gross monthlyincome (£000’s)

Observed frequency

Total monthlycost (£000’s)

Observed frequency

6 1 10 1

8 4 12 7

10 6 14 6

12 3 16 5

14 2 18 4

16 1 20 4

18 2 22 3

20 1

22 7

24 3

Income Frequency Probability Cumulative probability

Randomnumber

6 1 .03 .03 00–2

8 4 .13 .17 3–16

10 6 .20 .37 17–36

12 3 .10 .47 37–4614 2 .07 .53 47–52

16 1 .03 .57 53–56

18 2 .07 .63 57–62

20 1 .03 .67 63–66

22 7 .23 .90 67–89

24 3 .10 1.00 90–99

30 1.00


26/26


EXAMPLE 3 (continued)

PDD Monthly Cost

This completes parts (a) and (b). For (c), you need to describe how this process would berepeated over and over again, by computer, using, naturally, different random numbersequences every time. Then, summarised results would, it is argued, reflect reality quitewell – as long as initial model assumptions are deemed plausible and sensible.

Finally, for (d), some ‘hypothetical discussion’ can be given based on saying, ‘Suppose

the average of many simulations turned out to show, as in (c), a small profit of £2000 perannum for the company . . .’.

Cost Frequency Probability Cumulative probability

Randomnumber

10 1 .03 .03 00–2

12 7 .23 .27 3–26

14 6 .20 .47 27–46

16 5 .17 .63 47–62

18 4 .13 .77 63–76

20 4 .13 .90 77–89

22 3 .10 1.00 90–99

30 1.00

MonthSales Cost

Profit/LossRN RN

123456789101112

034743738636964736614698

81412222210241410181224

162277943949544354821737

121220221416161416201214

–42–808–680–6–2010

Σ = + 2

risk analysis and simulation 0066v1-2

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