Source: Project Management in Practice, 5th Edition, Mantel, Meredith, Shafer, Sutton, Wiley, 2014.

Project Management: A Systems Approach to Planning, Scheduling, and Controlling, 10th Edition, Harold Kerzner, Wiley, 2009.

Unit 6Planning and Budgeting the Project

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* It takes time.* You have to think.* It involves paper work.* You are bound to systematic procedures.* You are committed to achieve a specific

result within a specified time period.

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Inadequate planning is often the cause of problems during the life of a project.

“Ready, fire, aim” – insufficient planning

Careful planning is the key to project success but overdone planning could hurt.

“Paralysis by analysis” –overdone planning

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If the task is well understood prior to being performed, much of the work can be preplanned.

If the task is not understood, then during the actual task execution more knowledge is gained that, in turn, leads to changes in resource allocations, schedules, and priorities.

The more uncertain the task, the greater the amount of information that must be processed in order to ensure effective performance.

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To eliminate or reduce uncertainty To improve efficiency of the operation To obtain a better understanding of the

objectives To provide a basis for monitoring and

controlling work

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The primary function of a project plan is to serve the PM as a map of the route from project start to finish. It should contain 9 key elements.1. Overview brief description of project deliverables milestones expected profitability and competitive effects intended for senior management

2. Objectives detailed description of project’s scope and deliverables project mission statement

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3. General approach• technological and managerial approaches• relationship to other projects• deviations from standard practices

4. Contractual aspects• agreements with clients and third parties• reporting requirements• technical specifications• project review dates• compliance with legal/environmental constraints

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5. Schedules outline of all schedules and milestones

6. Resource requirements estimated project expenses overhead and fixed charges

7. Personnel• special skill requirements• necessary training• special legal arrangements (such as non-disclosure

agreements)

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8. Risk management listing of potential major and minor disasters late subcontractor deliveries, bad weather, unreasonable

deadlines, equipment failure, changes in project scope, etc.

Contigency plans to deal with the potential crisis.

9. Evaluation methods• evaluation procedures and standards• procedures for monitoring, collecting, and storing data

on project performance

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A Project Charter consists of most, if not all, of the nine items described in the project plan … plusthe sign-off on the plan by all major stakeholders

The signers include a representative of the project sponsor, the client user, the project manager, the program manager (if the project is part of an overall program) and other concerned stakeholders*Once agreed to, the charter cannot be altered by any signer without acceptance by the others.

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Review the project objectives with senior management – project scope plus expected desirable outcomes. It is for the PM to Understand the expectations that the organization,

the client, and other stakeholders have for the project.

Identify who among senior managers has a major interest in the project.

Determine if anything about the project is atypical.

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At least one representative from senior management (preferably the one with the strongest interest in the project, the project champion.)

Managers from functional areas that will contribute competencies or capacities to the project

[Perhaps] highly specialized technical experts

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Senior management introduces the project to the group, the project manager (to ensure that the group understands the firm’s commitment.)

Project manager chairs meeting• develop general understanding of the functional inputs

the project will need• may brainstorm the problem• may review a tentative budget and develop preliminary

plan Important results should include

• scope understood and temporarily fixed• functional managers understand their responsibilities

and have committed to developing the initial plan• potential benefits to the organization outside the scope

of the project are noted.

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The primary purpose here is to ensure that all tasks required to produce deliverables are accounted for (need to know what must be done, by whom, when with what resources.)

Hierarchical planning process – a systematic procedure.

• begin with project’s objectives• list major activities needed to achieve objectives (level 1 activities)• delegate level 1 activities to individuals or functional areas to

develop list of level 2 activities • degree of detail should be same within a given level

The outcome of this hierarchical planning is the development of a Work Breakdown Structure (WBS) for the project

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Gather project team Provide each member with a pad of Sticky-

Notes Define project’s objective and level 1

activities. Team members write down all the tasks

they can think of Sticky-Notes are then placed and arranged

on wall

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We are going to upgrade the network, workstations and software in an office.

We start by listing the high level tasks:– Order and install workstations– Order parts for network and install– Order and install software– Test upgrades

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Let’s show this as a diagram:– (The numbers represent the level of WBS)

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Is this enough detail? NO!

– What’s involved in ordering and installing hardware? Who does it? How long does it take?

– Similarly, what’s involved in ordering and installing network, software? Who does it? How long does it take?

– How do we test? Who does the testing?

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Let’s just take one item from our Level 1 list: Order and Install Hardware• It has at least 3 separate activities.

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Is this enough detail?– Will the same person do the work in each box?– Do we know how long each box will take to

complete?– Will each box take less than a week to complete?

If any of these answers is “No”, then we need to further decompose the task.

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Maybe Document Hardwae Requirements will take more planning and investigation:

– Someone will interview users to understand their needs.

– Someone will research hardware requirements to make sure the new machines are properly sized for the software.

– Somebody has to put this together into a specification.

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• We found the detail we need at WBS Level 3.• Now we do this for the rest of the WBS (you can do that on your own).

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►Microsoft Project (or MSP) is a project management software

►It is designed to assist project managers in:• developing plans• assigning resources to tasks• tracking progress• managing budgets• analyzing workloads

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Robust management tools with the right blend of:◦ Usability◦ Power◦ Flexibility

That’s why you can manage projects more efficiently and effectively.

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MSP enables you to stay informed and control:◦ Project work◦ Schedules◦ Finances

And to keep project teams aligned

Demo (from Scratch)

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Now that we have our WBS, we need to think about it a bit:

– What tasks must be completed first?– What tasks depend on other tasks to be done?– Are there tasks that must be done on a certain

date (and not before)?– Are there certain tasks that must be done by a

certain date (and not later)?

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A Dependency is a relationship where one task depends on another task/event before it can start or finish:

Dependencies come in 4 types:– Start-Finish– Start-Start– Finish-Start– Finish-Finish

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Start-Finish: I can only start to install the furniture when you are finished telling me where to put it. (most common)

Finish-Finish: I can only finish wiring the network when you finish installing the switches. (next common)

Start-Start: I can only start to carry this heavy box when you start to carry it with me.

Finish-Start: I can only finish my delivery when you start to unload the truck. (very uncommon)

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We can’t Draft Specifications until we are finished Interviewing Users and Gathering Software Requirements.

Therefore Draft Specifications has a Start-Finish dependency on these other two activities.

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Mandatory Dependencies are those that must be obeyed.– Eg. We can’t put the roof on a building until we finish the

walls. Discretionary Dependencies are “nice to have.” We

don’t normally pay attention to these.– Eg. Let’s wait until the computers arrive before we figure out

the furniture for them. External Dependencies are enforced by external

conditions.– Eg. The need for Parliament to pass the State Budget before

the project can start.

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Some tasks take a certain time to complete once started. Nothing can be done until the lag time is complete:– A baby has a 9 month lag time before you need to

buy diapers.– Cement has a lag time of a day or two before you can

walk on it.

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Lead Times are a “hurry up”. We can start on a task early but only if its preceding tasks are well underway.– Eg. It might take 4 weeks to plaster the whole house,

but the painter can start painting with a 2 week lead after the plastering starts. This is because the plasterer will finish one part of the house first before moving on to other walls.

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So we can’t just do activities in any old order. We need to figure out their dependencies (if any).

We figure out dependencies as a project team. All team members are involved in thinking through what needs to be done first.

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Develop a WBS for your project through a hierarchical planning process.

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The project WBS could serve as a project action plan if Project activities identified and arranged in

successively finer detail (i.e., by levels) Type and quantity of each required resource

identified for each activity Predecessors and task durations estimated for each

activity All project milestones identified Individual or group assigned to perform the work

identified for all activities

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The Project Master Schedule is created by combining milestones, task durations, and predecessors

This master schedule allows the project manager to compare actual and planned task durations and resource usage

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The RACI matrix is a tool used to help project managers better delegate tasks throughout a project and to stop or prevent project members doubling up on tasks and responsibilities. RACI is an acronym standing for

•Responsible•Accountable•Consulted•Informed

The RACI Matrix has project tasks in the rows and the departments/individuals in the columns.

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Mind mapping is a visual approach that closely mirrors the way the human brain records and stores information

In addition to its visual nature, this methodology has the advantage of tapping in to the creative potential of multiple team members

Mind mapping is an entertaining approach that helps generate enthusiasm and involvement

A fast and effective tool that can facilitate project planning and help minimize problems that result from inadequate upfront planning.

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Designate a team member as the facilitator. Taping a large sheet of paper on the wall. Begins by writing the project goal in the center. Team member brainstorm to identify the major

tasks that must be done to accomplish this goal. Continue break down components in the mind

map into more detailed tasks. In addition to words (key words), use picture and

images.

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Traditional project management tools (such as Gantt charts and precedence diagrams) tend to focus on those tasks that have to be completed in order for other to start

Often a more important question is: what information is needed from other tasks to complete specific task?

To address the issue of information flows, a design structure matrix could be used.

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a b c d e fab X Xc X Xd X X Xe Xf X X X

X -- information flow

• Identify all tasks and list them in the order they are typically carried out in both row and column.

• Place an X in the tasks in a row that supplied info to the task row.

For example-a and e supplies info to task b.B and f supplies info to task c.

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Participatory management style is institutionizedthrough Employee Involvement (EI), Six Sigma (SS), Quality Circle (QC), Total Quality Management (TQM), Self-Directed Work Team (SDWT), and Self-Managed Team (SMT).

Research suggests that the success of empowered teams depends heavily on how the team program is implemented rather than with the team itself.

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Teams generate high quality solutions Micromanagement is avoided Team is given accountability for part of the

project deliverable Synergistic solutions are frequent The PM has a tool for timely team

evaluation and feedback

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Budgets are plans for allocating organizational resources to project activities

• forecasting required resources, quantities needed, when needed, and costs

Budgets help tie project to overall organizational objectives

Budgets can be used as a tool by upper management to monitor and guide projects.

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The process of forecasting what resources the project will require, what quantities of each will be needed, when they will be needed, and how much they will cost. Direct cost could be estimated by

• Resource cost per unit (Table 4-1)• Resource assigned to different tasks. (Table 4-2)

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Table 4-1 Resource Cost per Unit for Producing a Short Documentary Film

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ID Resource Name Max. Units Std. Rate Ovt. Rate Cost/Use Accrue At

1 Scriptwriter 1 $75.00/hr $100.00/hr $0.00 Prorated

2 Producer 1 $100.00/hr $150.00/hr $0.00 Prorated

3 Client 0.2 $0.00/hr $0.00/hr $0.00 Prorated

4 Secretary 1 $25.00/hr $40.00/hr $0.00 Prorated

5 Editor 1 $50.00/hr $85.00/hr $0.00 Prorated

6 Production staff 1 $40.00/hr $70.00/hr $0.00 Prorated

7 Editing staff 1 $40.00/hr $70.00/hr $0.00 Prorated

8 Editing room 1 $0.00/hr $0.00/hr $250.00 Start

Table 4-2 Budget by Resource for Producing a Short Documentary Film

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ID Task Name Resource Work Hours Cost Task Duration

1 Project approval 0hr $0.00 0 days

2 Scriptwriting 112 hr $8,400.00 14 days

Scriptwriter 112 hr $8,400.00

3 Schedule shoots 240 hr $5,400.00 15 days

4 Begin scheduling 0hr $0.00 0 days

5 Propose shoots 120 hr $7,000 5 days

Scriptwriter 40hr $3,000.00

Producer 40hr $4,000.00

Client 40hr $0.00

6 Hire secretary 40 hr $4,000.00 5 days

Producer 40 hr $4,000.00

7 Schedule shoots 80hr $2,000.00 10 days

Secretary 80hr $2,000.00

Cost incurred PM – once a commitment was made using the

project’s resource. Accountant – once an invoice was received Controller – once a check for the invoice is

mailed.

* An accountant usually spreads a cost evenly over the project period, for example, an $8,000 charge in a four-month project will result in a $2,000 allocation per month.

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Based on collective judgments and experiences of top and middle managers

Overall project cost estimated by estimating the costs of major tasks

The estimates are then given to lower level managers to budget lower level tasks.

Advantages accuracy of estimating overall budget errors in funding small tasks need not be individually identified

Disadvantage Job always look easier, faster, and cheaper to the top management

than to the person who has to do them. Rely heavily on the experience and judgment of top management

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Work Breakdown Structure (WBS) identifies elemental tasks

Those responsible for executing these tasks estimate resource requirements.

All the estimated costs add up to the project budget.

Advantage more accurate in the detailed tasks Disadvantage risk of overlooking tasks “rare” in practice as budget is the upper management’s

primary tool for controlling the project.

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Determine resource requirements and then costs for each task (work element)• fixed costs (e.g., materials)• labor time and labor rate• equipment time and equipment rate• overhead• General, sales, and administrative (GS&A)

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Estimate the cost for a certain task• fixed costs – materials $235• labor time and labor rate – 16 hours at $10/hr• equipment time and equipment rate - none• Overhead – 50% of direct labor• General, sales, and administrative (GS&A) –

40% of all of the above

Cost = {$235 + [(16 x $10/hr) x 1.5]} x 1.4= $665

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Type 1 (S shaped) – a somewhat reduced level of resource will cause a smaller than proportional cut in the project’s objectives or performance. (not a big problem)

Type 2 (J shaped) – a somewhat reduced level of resource will cause a Larger than proportional cut in the project’s objectives or performance. (a serious problem)

Activity oriented budgeting are based on historical data accumulated through an activity-based accounting system

• Show lines of standard activity by actual use and budget for given time period. (Table 4-3)

With Program budgets, each project is divided by task and time period

• Show expenses by task and time period. (Table 4-4)

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Table 4-3 Typical Monthly Budget for a Real Estate Project

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Current

Actual Budget Variance Pct.

Corporate-Income Statement

Revenue 8430 Management fees .00 .00 .00 .0 8491 Prtnsp. reimb.-property mgmt. 7,410.00 6,222.00 1,188.00 119.0 8492 Prtnsp. reimb.-owner acquisition

.00 3,750.00 3,750.00- .0

8493 Prtnsp. reimb.-rehab .00 .00 .00 .0 8494 Other income .00 .00 .00 .0 8495 Reimbursements-others .00 .00 .00 .0

Total revenue 7,410.00 9,972.00 2,562.00- 74.3

Operating expenses

Payroll & PIR benefits 8511 Salaries 29,425.75 34,583.00 5,157.25 85.0 8512 Payroll taxes 1,789.88 3,458.00 1,668.12 51.7 8513 Group ins. & med. reimb. 1,407.45 1,040.00 387.45- 135.3 8515 Workers' compensation 43.04 43.00 .04- 100.0 8516 Staff apartments .00 .00 .00 .0 8517 Bonus .00 .00 .00 .0

Total payroll & PIR benefits 32,668.12 39,124.00 6,457.88 83.5

Table 4-4 Project Budget by Task and Month

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Start End Estimate Monthly Budget (£) Task Node (I) Node (J) (£) 1 2 3 4 S 6 7 8

A 1 2 7000 5600 1400

B 2 3 9000 3857 5143

C 2 4 10000 3750 5000 1250

D 2 5 6000 3600 2400

E 3 7 12000 4800 4800 2400

F 4 7 3000 3000

G 5 6 9000 2571 5143 1286

H 6 7 5000 3750 1250

I 7 8 8000 2667 5333

J 8 9 6000 6000 -- --

75000 5600 12,607 15,114 14,193 9836 6317 5333 6000

“Each person who has a task to do must own the design and execution of that task and must be held

responsible for its timely achievement.”“Accountability is the twin of empowerment.”

By Jim McCarthy, Dynamics of Software Development , Microsoft Press, 1995

Forms Learning curves Tracking signals

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A form for gathering data on project resource needs might include:

• people – managers, technical and non-technical• money• materials – facilities, equipment, tools, space, etc.• special services

And might identify:• person to contact• how many/much needed• when needed• whether available

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rn nTT 1

where

Tn = the time required to complete the nth unit

T1 = the time required to complete the first unit

r = log(learning rate)/log(2)

*Learning rate is typically between 70 and 95%

Suppose that a company is to produce 10 units of a complex device, one at a time. The job has never been done before. The labor, cost, and time needed for the first unit will be much higher than the remaining unit due to the learning effect.

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Example – Suppose that Media One Consultants is developing multimedia teaching materials for an 18-chapter bookwhereT1 = 21 hoursLearning rate = 78%

The time to complete the2nd chapter, T2 = 21 x 2 log(0.78)/log(2) = 16.38 hrs;3rd chapter, T3 = 21 x 3 log(0.78)/log(2) = 14.16 hrs;

See Excel Example73

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Media One Consultants

Learning Rate 78%Hourly Rate $65

Overhead Rate 20%

Time for 1st Unit 21

With Learning

Curve Without Learning Curve

Cumulative Chapter Cumulative Cumulative Chapter Cumulative

Chapter # Unit Time Time Cost Cost Chapter # Unit Time Time Cost Cost1 21.00 21.00 $1,638.00 $1,638.00 1 21.00 21.00 $1,638.00 $1,638.002 16.38 37.38 $1,277.64 $2,915.64 2 21.00 42.00 $1,638.00 $3,276.003 14.16 51.54 $1,104.81 $4,020.45 3 21.00 63.00 $1,638.00 $4,914.004 12.78 64.32 $996.56 $5,017.01 4 21.00 84.00 $1,638.00 $6,552.005 11.79 76.11 $919.95 $5,936.96 5 21.00 105.00 $1,638.00 $8,190.006 11.05 87.16 $861.75 $6,798.71 6 21.00 126.00 $1,638.00 $9,828.007 10.45 97.62 $815.43 $7,614.14 7 21.00 147.00 $1,638.00 $11,466.008 9.97 107.58 $777.32 $8,391.46 8 21.00 168.00 $1,638.00 $13,104.009 9.55 117.14 $745.18 $9,136.64 9 21.00 189.00 $1,638.00 $14,742.00

10 9.20 126.34 $717.56 $9,854.20 10 21.00 210.00 $1,638.00 $16,380.0011 8.89 135.23 $693.46 $10,547.66 11 21.00 231.00 $1,638.00 $18,018.0012 8.62 143.84 $672.17 $11,219.83 12 21.00 252.00 $1,638.00 $19,656.0013 8.37 152.22 $653.16 $11,872.99 13 21.00 273.00 $1,638.00 $21,294.0014 8.15 160.37 $636.03 $12,509.02 14 21.00 294.00 $1,638.00 $22,932.0015 7.96 168.33 $620.50 $13,129.52 15 21.00 315.00 $1,638.00 $24,570.0016 7.77 176.10 $606.31 $13,735.82 16 21.00 336.00 $1,638.00 $26,208.0017 7.61 183.71 $593.27 $14,329.10 17 21.00 357.00 $1,638.00 $27,846.0018 7.45 191.16 $581.24 $14,910.34 18 21.00 378.00 $1,638.00 $29,484.00

Two types of error – random error and biased error. The former is by chance, cancelled out; the latter is not by chance could not cancel out.

A tracking signal number can reveal if there is a systematic bias in cost and other estimates … and whether the bias is positive or negative

By observing their own errors, a project manager can learn to make unbiased estimates

See Excel example

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Figure 4-3

TrackingPeriod (t)Estimate Actual (At/Et) - 1 |(At/Et) - 1| MAR* Signal**

1 155 163 0.0516 0.05162 242 240 -0.0083 0.0083 0.0299 1.44793 46 67 0.4565 0.4565 0.1721 2.90404 69 78 0.1304 0.1304 0.1617 3.89785 75 71 -0.0533 0.0533 0.1400 4.12026 344 423 0.2297 0.2297 0.1550 5.20507 56 49 -0.1250 0.1250 0.1507 4.52348 128 157 0.2266 0.2266 0.1602 5.6700

Total 0.9082 1.2814

* MARi = [Sum (i = 1 to t)(|(Ai/Ei) - 1|)]/i** Tracking Signali = [Sum (i = 1 to t)((Ai/Ei) - 1)]/MARi

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> 0 underestimate, < 0 overestimate

Small tracking signal => little or no biasTracking signal > 1 => Positive biasTracking signal < -1 => negative bias

Changes in resource prices• estimate rate of price change individually for

inputs that have significant impact on costs Waste and spoilage Team member turnover costs Using “mythical man-months” Organization climate Just bad luck

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Errors made by cost estimator as to how to achieve tasks.

New knowledge about the nature of the performance goal or setting.

A mandate … a new law or standard, etc.

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Accept a negative change and take a loss on the project (least preferred)

Prepare for change ahead of time … include provisions in the original contract that allow for renegotiated price and schedule for client-ordered changes in performance (best practice)

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Risk management focuses on:◦ Known unknowns◦ Proactive management◦ The future.

Historically, we focused on schedule and cost risk management.

Today, our focus is on technological risk management:◦ Can we design and build it?◦ What is the risk of obsolescence?

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Risk = f(Likelihood, Impact)

•Likelihood is the probability of occurrence•Impact is the amount at stake

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•External – unpredictable

•External – predictable

•Internal – non-technical

•Internal – technical

•LegalCategories of Risks (Boeing)

Financial risks

Market risks

Technical risks

Production risks

Risk management planning Risk identification Qualitative risk analysis Quantitative risk analysis Risk response planning Risk monitoring and control

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STAGE I STAGE II GUID-ANCE

WARHEAD

COST

MANU.

TEST

DESIGN

HIGH

MEDIUM

LOW

LEGEND

PROGRAMSUMMARY

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Customer’sKnowledgeCustomer’sKnowledge

ExperiencedExperienced

InexperiencedInexperienced

SimpleSimple ComplexComplex

Contract TypeContract Type

List ways project might potentially fail Evaluate severity (S) of each failure “1” represents failure with no effect and “10”

represents very severe and hazardous failure Estimate likelihood (L) of each failure

occurring “1” indicating that failure is rather remote and not

likely to occur and “10” indicating that failure is almost certain to occur

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Estimate ability to detect each failure (D)• “1” is used when monitoring and control systems are

almost certain to detect the failure and “10” where it is virtually certain the failure will not be detected

Calculate Risk Priority Number (RPN)• Multiply S, L and D together Sort potential failures by their RPNs

• Consider ways of reducing the risk associated with failures with high RPNs

(Table 4-5)

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Failure S L D RPN*Not effective 8 6 5 240Not safe 8 4 5 160Drug interacts with other drugs

6 3 8 144

Beat to market (by competitor)

7 3 2 42

Can't produce 6 4 4 96RPN (Risk Priority Number) = S x L x D

In this example, the most significant risk is that the new drug is not effective.

Table 4-5 FMEA for New Product Development Project at Pharmaceutical Company

Assume that all possible costs for some tasks could be represented by the beta distribution as shown in figure 5-13 The Expected Cost (EC) is based on three

estimates• pessimistic (a) – the actual cost of the task will

be a or higher less than 1 percent of the time• most likely (m) – the mode of the distribution• optimistic (b) – the actual cost of the task will be

b or lower less than 1 percent of the time

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𝐸𝐶𝑎 4𝑚 𝑏

6

Figure 5‐13 The Beta Distribution of all Possible Times for an Activity

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The Beta Distribution

Table 4-6 Optimistic, Normal, and Pessimistic Cost Estimates for the Annual Tribute Dinner

Budget InformationAnnual Tribute Dinner

Optimistic Normal Pessimistic Expected Cost Task Name Cost=a Cost=m Cost=b =(a+4m+b)/6 Begin preparations for tribute dinner Select date & secure room Obtain corporate sponsorships for event $100.00 $150.00 $350.00 $175.00 Identify potential businesses to sponsor Phone/write businesses $100.00 $150.00 $350.00 $175.00

Event hosts/Me Identify and secure honoree of event Identify and secure master of ceremonies Identify/secure person to introduce honoree Identify/secure event hosts & hostesses

Invitations

Secure mailing lists Design invitation with PR firm $1,250.00 $1,500.00 $2,200.00 $1,575.00

Print invitation $2,300.00 $2,500.00 $3,000.00 $2,550.00

Mail invitation $250.00 $300.00 $410.00 $310.00

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RSVP's back Event entertainment secured $750.00 $1,000.00 $1,250.00 $1,000.00

Food and drink

Finalize menu

Identify menu options

Trial menus

Select final menu $700.00 $750.00 $800.00 $750.00 Identify company to donate wine Table decorations, gifts, cards, flowers, etc. $2,200.00 $2,800.00 $3,250.00 $2,775.00

Identify and have made event gift to attendees $2,000.00 $2,500.00 $3,000.00 $2,500.00 Find florist to donate table arrangements Hire calligrapher to make seating cards $200.00 $300.00 $400.00 $300.00

Develop PR exhibit to display at event $75.00 $150.00 $225.00 $150.00 Event and honoree publicity $200.00 $325.00 $450.00 $325.00

Hire event photographer $400.00 $450.00 $500.00 $450.00

Finalize seating chart

Hold tribute dinner Send out "thank you's" to sponsors & donators $75.00 $150.00 $225.00 $150.00

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When probability information is available for a cost information, the expected value of an activity cost is a preferred estimate.

Suppose that the cost of an activity (x) has the following probability distribution:

x $100 110 120 130 140f(x) 0.2 0.1 0.3 0.3 0.1

Find the expected value of X.

E(X) = 100·0.2 + 110·0.1 + 120·0.3 + 130·0.3 + 140·0.1 = 124

*Compare to the (a + 4m +b) /6 = (100 + 4x120 + 140)/6 = 120

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Game theory Assume that competitors and the environment

are your enemies Select a course of action that minimizes the

maximum harm Simulation Use probabilistic information associated with

each activity Employ Monte-Carlo simulation techniques to

estimate the likelihood that a certain risk might occur.

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‐ The End ‐

Thank You!

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