GENERAL_ASPECTS_OF_ENERGY_MANAGEMENT_AND_ENERGY_AUDIT (CHAPTER 8:PROJECT MANAGEMENT)

 

 GENERAL_ASPECTS_OF_ENERGY_MANAGEMENT_AND_ENERGY_AUDIT  

(CHAPTER 8:PROJECT MANAGEMENT)

What is a Project?

A project is a “temporary endeavor undertaken to create a unique product or service”.

Projects are temporary because they have a definite beginning and a definite end. They are unique because the product or service they create is different in some distinguishing way from similar  products or services.

Designing, installation and commissioning of a cogeneration system in an industry is an example of a project. The unique work is defined by the cogeneration system and has a specific beginning and end. A project manager is responsible for the project, overseeing the contractors and managing the schedule and budget.

Projects are means for change. Such change can be sought or it can be resisted. Changing practices or investing money for energy efficiency is an occurrence of change. Fortunately, energy related projects generally have low risk attached to both the means of achieving change and result of that change. The energy manager should seek opportunities to save energy and aim to influence all projects in which energy is a significant factor. It may be that, in the early stages, an energy benefit will add to the arguments in favour of the project. Investment in energy efficiency is also good publicity news that can be given to customers and media.

Project management is a set of principles, methods, and techniques that people use to effectively plan and control project work. Project management principles and techniques help complete projects on schedule, within budget, and in full conformance to project specifications. At the same time, they help achieve the other goals of the organization, such as productivity, quality, and cost effectiveness.

The objective of project management is to optimize project cost, time, and quality.

Project Development Cycle (PDC)

The various steps in the PDC are:

1. Project Identification and Screening

2. Technical Design

3. Financing

4. Contracting

5. Implementation

6. Performance Monitoring

Project Identification and Screening

The first step in the project development cycle is to identify components of the project. Projects may be identified both internally and externally:

1.Internal identification takes place when the facility manger identifies a package of energy saving opportunities during the day-to-day energy management activities, or from detailed energy audits.

2.External identification of energy savings can occur through systematic energy audits undertaken by areputable energy management consultant, energy service company or industry organization.

In screening projects, the following criteria should be used to rank project opportunities:

1.Economic feasibility of energy savings measures (Internal rate of return, net present value, cash flow, average payback)

2.Sustainability of the savings over the life of the equipment.

3.Ease of quantifying, monitoring, and verifying energy savings.

4.Availability of technology, and ease of adaptability of the technology to Indian conditions.

5.Other environmental and social cost benefits (such as reduction in GHG emissions and local pollutants such as SO. emissions)

Technical Design

For a project to be considered a viable investment, the project proponent must present a sound technical feasibility study that identifies the following elements in detail:

* The proposed new technologies, process modifications, equipment replacements and other measures included in the project.

* Product/technology/material supply chain (e.g., locally available, imported, reliability of supply)

* Any special technical difficulties (installation, maintenance, repair), associated skills required.

* Preliminary designs, including schematics, for all major equipment needed, along with design requirements, manufacturer’s name and contact details, and capital cost estimate.

* Organizational and management plan for implementation, including timetable, personnel requirements, staff training, project engineering, and other logistical issues.

Financing

When considering a new project, it should be remembered that other departments in the organization would be competing for capital for their projects. However, it is also important to realize that energy efficiency is a major consideration in all types of projects, whether they are:

1.Projects designed to improve energy efficiency

2.Projects where energy efficiency is not the main objective, but still plays a vital role.

Most organization reaches the point when all the obvious measures to save energy have been taken and capital investment is needed to make further savings. Low cost measures for saving energy, which can be treated as mini projects in their own rights, should be given top priority. It is necessary to ensure that the present system is operating efficiently before spending any money.

If all the projects in a portfolio of applications to top management are within the policy and the procedures of the organization, then the one chosen for approval is likely to be the one, which gives the best return on the investment, and which is best presented. Many organizations have a priority list which gives preference to capital expenditure on projects offering certain advantages or removing particular disadvantages.

Project funds can be obtained from either internal or external sources.

Internal sources include:

Direct cash provision from company reserves

¢ Revenue budget (if payback is less than one year)

¢ New share capital

External sources of funds include:

¢ Bank loans

¢ Leasing arrangement

¢ Payment by savings i.e. a deal arranged with equipment supplier

e Energy services contract

¢ Private finance initiative

Before applying for project fund, all the options for funding the project are discussed with finance managers. Often, energy savings improve viability of non-energy projects.

The availability of external funds depends on the nature of your organization. The finance charges on the money borrowed will have an impact on the validity of the project.

If outside financing is sought for an energy management project, it may be obtained from a private bank, or from one of the special financing programs offered by national development banks and other funding agencies. In addition to the usual information on company assets and lines of credit, financial agencies will require an assessment of the financial feasibility of the proposed project. This should include a fully specified pro forma financial worksheet that presents project cash flows, net present value, and internal rate of return.

Contracting

Since a substantial portion of a project is typically executed through contracts, the proper management of contracts is critical to the successful implementation of the project. In this context, the following should be taken care.

1.The competence and capability of all the contractors must be ensured-one weak link can affect the timely performance of the contract.

2.Proper discipline must be inculcated among contractors and suppliers by insisting that they should develop realistic and detailed resource and time plans that are matching with the project plan.

3. Penalties-which may be graduated-must be imposed for failure to meet contractual obligations. Likewise, incentives may be offered for good performance.

4. Help should be extended to contractors and suppliers when they have genuine problems-they should be regarded as partners in a common pursuit.

5. Project authorities must retain independence to off-load contracts (partially or wholly) to other parties well in time where delays are anticipated.

If the project is to be implemented by an outside contractor, several types of contract may be used to undertake the installation and commissioning:

Traditional Contract: All project specifications are provided to a contractor who buys and installs equipment at fixed price or cost plus a mark-up. Also called as fixed-price or lump sum contract, the contractor takes the risk of unforeseen problems in exchange for a larger profit. This type of contract is appropriate when dealing with unknown vendors or when project manager anticipates the work is risky.

Extended Technical Guarantee/Service: The contract offers extended guarantees on the performance of selected equipment, and/or offers service/maintenance agreements.

Extended Financing Terms: The contractor provides the option of an extended lease or other financing vehicle in which the payment schedule can be based on the expected savings.

Guaranteed Saving Performance Contract: All or part of savings is guaranteed by the contractor, and all or part of the costs of equipment and/or services is paid down out of savings as they are achieved.

Shared Savings Performance Contract: The contractor provides the financing and is paid an agreed fraction of actual savings as they are achieved. This payment is used to pay down the debt costs of equipment and/or services.

Implementation

A great deal of the emphasis in the planning stage of any project is on understanding where and when problems may occur. Many projects introduced by energy managers end up as some other manager’s responsibility, e.g. a production manager or a works engineer. The following needs to be thought ahead and anticipated.

¢ Type and extent of measurements needed to control and measure the success of the project

¢ Winning the confidence and cooperation of key personnel involved.

¢ Timely and frequent communication between participants.

With proper techniques, changes and modifications in project can be understood and incorporated without loss of control. Before considering the components of a plan, its purpose must be defined. A plan turns a proposed project into reality. As reality often differs from theory, the plan should consider as many technical, financial and other ‘what ifs’ as possible.

Performance Monitoring

In order to keep a tab on the progress of the project, a system of monitoring must be established. This helps in:

¢ Anticipating deviations from the implementation plan

¢ Analyzing emerging problems

¢ Taking corrective action

In developing a system of monitoring, the following points must be borne in mind:

¢ It should focus sharply on the critical aspects of project implementation.

¢ It must lay more emphasis on physical milestones and not on financial targets.

¢ Monitoring must be kept simple.

Project Review

Once the project is commissioned the review phase has to be set in motion. Performance review should be done periodically to compare actual performance with projected performance. A feedback device, it is useful in several ways:

a) It throws light on how realistic were the assumptions underlying the project

b) It provides a documented log of experience that is highly valuable in future decision making

c) It suggests corrective action to be taken in the light of actual performance

d) It helps in uncovering judgmental biases

Project Planning Techniques

To achieve control of a project, it is necessary to plan. For more complex projects, the more advanced

planning methods are needed. Many project management software packages are available with detailed

operational manuals.

Work Breakdown Structure (WBS)

Work Breakdown Structure (WBS) is the process of dividing complex projects to simpler and manageable tasks. The project managers use this method for simplifying the project execution. In WBS, much larger tasks are broken down to manageable chunks of work. These chunks can be easily supervised and estimated.

First main deliverable of a project is identified and then the higher levels tasks are broken into smaller chunks of work. In the process of breaking down the tasks, one can break them down into different levels of detail. The level of breakdown detail depends upon project type and the management style followed for the project. WBS can be displayed using tree structure or list or tables.

WBS is developed before dependencies are identified and activity durations are estimated. The WBS can be used to identify the tasks before constructing Gantt chart and networks such as Critical Path Method -CPM and Program Evaluation and Review Technique- PERT.

Gantt chart

During the era of scientific management, Henry Gantt developed a tool for displaying the progress of a project in the form of a specialized chart. An early application was the tracking of the progress of ship building projects. Today, Gantt’s scheduling tool takes the form of a horizontal bar graph and is known as a Gantt chart.

Gantt chart is now commonly used for scheduling the tasks and tracking the progress of energy management projects. Gantt charts are developed using bars to represent each task. The length of the bar shows how long the task is expected to take to complete. The duration is easily shown on Gantt charts.

A basic sample for a pumping station is shown in Figure 8.1. The horizontal axis of the Gantt chart is a time scale, expressed either in absolute time or in relative time referenced to the beginning of the project. The time resolution depends on the project - the time unit typically is in weeks or months. Rows of bars in the chart show the beginning and ending dates of the individual tasks in the project.

In the example in Figure 8.1, each task is shown to begin when the task preceding it completes. However, the bars may overlap in cases where a task can begin before the completion of another, and there may be several tasks performed in parallel. For such cases, the Gantt chart is quite useful for communicating the timing of the various tasks.

Limitation of Gantt chart

The Gantt chart does not normally show the logical interdependencies between the predecessor and successor activities very well. Such requirements are best served by the network diagram, which shows logic clearly but does not have a time scale axis like the Gantt chart.

Gantt Chart Enhancement

This basic version of the Gantt chart often is enhanced to communicate more information.

 1.A vertical marker can used to mark the present point in time (If a vertical line is drawn through a given day, the activities that are scheduled to be taking place simultaneously will have the line pass through them. In this way the need for simultaneous resources can clearly be seen).

2.The progression of each activity may be shown by shading the bar as progress is made, allowing the status of each activity to be known with just a glance.

3. Dependencies can be depicted using link lines or color codes.

4.Resource allocation can be specified for each task.

5. Milestones can be shown.

Project Networking Techniques

Project network shows dependency relationships between tasks/activities in a project in a graphical view. It shows clearly tasks that must precede or succeed other tasks in a logical manner. It is a powerful tool for planning and controlling project

Network Definitions

Activity: Any portions of project (tasks) which required by project, uses up resource and consumes time.

Event: Beginning or ending points of one or more activities are called ‘nodes’

Network: Combination of all project activities and the events

Two ways of representing activities are activity-on-arrow (AOA) and activity-on-node (AON).

Activity-on-arrow (AOA) is represented by the activities on arrows which are connected at events (nodes shown as circles) to show the dependencies (preceding or succeeding) as shown in Figure 8.2.

Activity-on-node (AON) is represented as nodes on the network and events that signify the beginning or ending of activities are depicted as arcs or lines between the nodes as shown in Figure 8.3. This method is used in project management software packages.

Dummy activity is required if two or more activities having identical starting and ending events. In above Figures, it is shown as dotted line to ensure that activity C starts only after activity B and activity D are completed.

Critical Path Method (CPM)

In 1957, DuPont developed a project management method designed to address the challenge of shutting down chemical plants for maintenance and then restarting the plants once the maintenance had been completed. Given the complexity of the process, they developed the Critical Path Method (CPM)

for managing such projects.

CPM provides the following benefits:

¢ Provides a graphical view of the project.

¢ Predicts the time required to complete the project.

¢ Shows which activities are critical to maintaining the schedule and which are not.

Steps in CPM Project Planning

1.Specify the individual activities.

2.Determine the sequence of those activities.

3.Draw a network diagram.

4.Estimate the completion time for each activity.

5.Identify the critical path (longest path through the network)

6.Update the CPM diagram as the project progresses.

1. Specify the Individual Activities

From the work breakdown structure, list the activities in the project. The project (made up of several tasks) should have only a single start activity and a single finish activity.

2. Determine the Sequence of the Activities

Develop the relationship among the activities. Decide which activities must precede and which must follow others.

3. Draw the Network Diagram

Draw the “Network” once all the activities and their sequencing have been defined. Network diagram originally was developed as an activity on node (AON) network, but some project planners prefer to specify the activities on the arrow.

4. Estimate Activity Completion Time

Assign time and/or cost estimates to each activity using past experience or the estimates of knowledgeable persons.

CPM is a deterministic model that does not take into account variation in the completion time, so one fixed time is used for an activity.

5. Identify the Critical Path

The critical path is the longest-duration path through the network. The significance of the critical path is that the activities that lie on it cannot be delayed without delaying the project. Because of its impact on the entire project, critical path analysis is an important aspect of project planning.

The critical path can be identified by determining the following four parameters for each activity:

1.ES - Earliest start time: the earliest time at which the activity can start given that its precedent activities must be completed first.

2. EF - Earliest finish time: equal to the earliest start time for the activity plus the time required to complete the activity.

3. LF - Latest finish time: the latest time at which the activity can be completed without delaying the project.

4.LS - Latest start time, equal to the latest finish time minus the time required to complete the activity.

The total float (slack time) for an activity is the time between its earliest and latest start time, or between its earliest and latest finish time. Slack is the amount of time that an activity can be delayed past its earliest start or earliest finish without delaying the project.

The critical path is the path through the project network in which none of the activities have slack, that is, the path for which ES=LS and EF=LF for all activities in the path. A delay in the critical path delays the project. Similarly, to accelerate the project it is necessary to reduce the total time required for the activities in the critical path.

6. Update CPM Diagram

As the project progresses, the actual task completion times will be known and the network diagram can be updated to include this information. A new critical path may emerge, and structural changes may be made in the network if project requirements change.

Example 8.1: Illustration of CPM

WBS of a project is given in Table 8.2.

The information in WBS indicates that the total time required to complete activities is 51 weeks. However, it can be seen from the network that several of the activities can be conducted simultaneously (A and B, for example). It is necessary to calculate critical path of the network.

Earliest Start Time and Earliest Finish Time

We can do a forward pass calculations. Starting at the network’s origin (node 1) and using a starting time of 0, we compute an earliest start (ES) and earliest finish (EF) time for each activity in the network. The expression EF = ES + t can be used to find the earliest finish time for a given activity. The completion time or the duration of an activity is represented as t. 

For example, for activity A, ES = 0 and t = 5; thus the earliest finish time for activity Ais EF =0+ 5 = 5. Refer Figure 8.4.

Completing the network from start based on above (Forward pass), we can calculate all ES and EF

time as shown in Figure 8.5.

Similarly, we can do a backward pass calculation. Starting at the completion point (node 7) and using a latest finish time (LF) of 26 for activity I, we trace back through the network computing a latest start (LS) and latest finish time for each activity

The expression LS = LF - t can be used to calculate latest start time for each activity. For example, for activity I, LF = 26 and t = 2, thus the latest start time for activity I is LS = 26 — 2 = 24.

Float or Slack or Free Time

Slack is the length of time an activity can be delayed without affecting the completion date for the entire project. For example, slack for C = 3 weeks, 1.e Activity C can be delayed up to 3 weeks {LS (8 weeks) — ES (5 weeks) or LF (12 weeks) — EF (9 weeks) = 3 weeks} as given in Figure 8.6.

Float or slack shows how much allowance each activity has i.e. how long it can be delayed without affecting the completion date of the project. Critical path is a sequence of activities from start to finish with zero slack. Critical activities are activities on the critical path.

Critical path identifies the minimum time to complete project. If any activity on the critical path is shortened or extended, project time will be shortened or extended accordingly. If resources have to be spent to speed up some activities, it should be only for critical activities. Such shortening or project in critical path by adding resources is called project crashing. If resources have to be saved by lengthening some activities, it should be only for non-critical activities, up to the limit of float.

The total time to complete the activities is 26 weeks. The activities which are critical and must be completed to keep the project within schedule are A, E, F, G, and I. Non-critical activities can be delayed before they cause delay in project completion time as per float or slack shown against each activity shown in Table 8.3.

Program Evaluation and Review Technique (PERT)

The Program Evaluation and Review Technique (PERT) 1s a probabilistic network model that allows for randomness in activity completion times. PERT was developed in the late 1950’s for the U.S. Navy’s Polaris project having thousands of contractors. It has the potential to reduce both the time and cost required to complete a project.

PERT steps are similar to CPM with only difference as to the estimation of time. Unlike CPM where times can be estimated with relative certainty, confidence, PERT uses 3 time estimates.

Tm = most likely time estimate

To= optimistic time estimate

Tp= pessimistic time estimate

PERT assumes a normal bell-shaped distribution (beta probability distribution) for the time estimates. For a beta distribution, the expected time for each activity can be approximated using the following weighted average:

Expected time = (Optimistic + 4 x Most likely + Pessimistic) / 6

The variance in the project completion time can be calculated by summing the variances in the completion times of the activities in the critical path. Given this variance, one can calculate the probability that the project will be completed by a certain date assuming a normal probability  distribution for the critical path. The normal distribution assumption holds if the number of activities in the path is large enough.

The calculation procedure is similar to CPM. The expected times may be displayed on the network diagram. The various times are calculated using the expected time for the relevant activities. The earliest start and finish times of each activity are determined by working forward through the network and determining the earliest time at which an activity can start and finish considering its predecessor activities.

The latest start and finish times are the latest times that an activity can start and finish without delaying the project. LS and LF are found by working backward through the network. The difference in the latest and earliest finish or latest and earliest start of each activity is that activity’s slack. The critical path then is the path through the network in which none of the activities have slack.

Example 8.2: Illustration of PERT

For the following project details (Table 8.4), PERT chart has to be prepared.

Table 8.4 Project Details

PERT Network diagram showing activities, ES, EF in forward pass is shown in Figure 8.7.

PERT Network diagram showing LS, LF in backward pass is shown in the Figure 8.8.

The details of network calculations from forward and backward pass are summarized in Table 8.5

Such PERT chart is not final. It may be necessary to make adjustment in PERT chart as project progresses. As the project unfolds, the estimated times can be replaced with actual times. In cases where there are delays, additional resources may be needed to stay on schedule and the PERT chart may be modified to reflect the new situation.

Benefits of PERT

PERT is useful because it provides the following information:

1.Expected project completion time.

2.Probability of completion before a specified date.

3.The critical path activities that directly impact the completion time.

4.The activities that have slack time and that can lend resources to critical path activities.

5.Activities start and end dates.

Limitations of PERT

1.The activity time estimates are somewhat subjective and depend on judgment.

2.Even if the activity times are well-estimated, PERT assumes a beta distribution for these time estimates, but the actual distribution may be different.

3.Even if the beta distribution assumption holds, PERT assumes that the probability distribution of the project completion time is the same as that of the critical path. Because other paths can become the critical path if their associated activities are delayed, PERT consistently underestimates the expected project completion time.

Implementation Plan for Top Management

Implementation is the stage where all the planned activities are put into action. Before the implementation of a project, the top management should identify their strength and weaknesses (internal forces), opportunities and threats (external forces).

The strength and opportunities are positive forces that should be exploited to efficiently implement a project. The weaknesses and threats are hindrances that can hamper project implementation. The implementers should ensure that they devise means of overcoming them.

Monitoring is important at this implementation phase to ensure that the project is implemented as per the schedule. This is a continuous process that should be put in place before project implementation

starts.

As such, the monitoring activities should appear on the work plan and should involve all stake holders. If activities are not going on well, arrangements should be made to identify the problem so that they can be corrected. When implementation of the project is not on target, there is a need for the project managers to ask themselves and answer the question, “How best do we get there?”

The implementation plan should include a project schedule or timeline. The level of detail included in a schedule depends upon the role of individual on the project. The Table 8.6 shows an example of high-level schedule that was developed by boiler engineer.

Planning Budget

The project budget will be used throughout the remainder of the project to track project, expenses and measure the money actually spent on project activities against the estimates given for those activities. The final budget figures are based on estimates provided by the project team, key stakeholders,  endors, and others after careful review of the Planning documents.

Most of the costs expended on the project, known as project costs, are fairly obvious and apply to most projects. For example, salaries, office supplies, and telephone charges will apply to almost all projects. These project costs and most others fall into one of three categories: human resource costs,  administrative costs, or resource costs.

Human Resource Costs are the costs associated with the personnel on the project. They include salaries

and the cost of benefits (such as vacation rime and health insurance).

Administrative costs are the everyday costs that support the work of the project but are not necessarily directly related to a specific task on the project. Local telephone expense, copier paper, heating  expenses, and support personnel are examples of administrative costs.

Resource costs include things such as materials needed for specific tasks, equipment leases, longdistance telephone expense, travel expenses, and so on. These expenses are specific to the project.

Direct Costs versus Indirect Costs

In addition to the three categories of project costs, there are also two types of costs: direct and indirect costs. Direct costs include costs such as salaries, equipment rentals, and training for team members.

Any cost that can be directly attributed to project work is a direct cost.

Indirect costs are not specific to the project. For example, project team may work with other members

of the organization (who are not working on the project) in the same building. The lease cost for the

building is an example of an indirect cost because it is not specifically related to the project since all

the company employees work there.

Another example of an indirect cost is administrative staff, managers, or other functional members

who will be assisting the project team with project activities (such as someone from the procurement

department) but are not assigned tasks themselves. Each organisation has its own procedures for

accounting for indirect costs. Finance or Accounting Department can provide guidance on how indirect

cost should be accounted for in the project budget.

Budget Estimation

If the project ends up as an unsuccessful project, one of the reasons could be inadequate budget. The responsibility for determining project cost lies mostly with finance manager and he gets approval from senior management. The project manager is told about the final budget and is expected to work with the allocated budget. If this is the case, project manager can contribute to budgeting process by using his communication and negotiation skills with finance manager.

The first step in creating the project budget is to gather all the Planning documents for reference. The project goals and deliverables for obvious budget expenses have to be reviewed. The WBS, network diagram also has to be reviewed to identify costs associated with them.

The WBS is used to identify, estimate, and assign costs to each element of the project. The work package level is where individual costs and estimates for each task (provided the work package level shows tasks) are shown, whereas the higher levels in the WBS will show rolled-up total costs. Analogous Estimating establishes an estimate for the current project based on the actual costs of previous projects that are similar in size and scope to the ongoing project. Bottom-up Estimating establishes individual estimates for each tasks and adds them all together to determine the total  estimate for the project.

Procurement Procedures

Many organizations have a procurement department that will help you purchase the items needed for your protect. They may also help you create the budget. Always check with the procurement  department to determine whether there are procedures you should follow when preparing your budget. Also be certain to work with the procurement department when purchasing services from vendors or contractors.

Construction

Project construction is the process of assuring that all systems and components of an industrial equipment are designed, installed, tested, operated, and maintained according to the operational requirements of the user. A commissioning process may be applied not only to new projects but also to existing units and systems subject to expansion, renovation or revamping.

In practice, the commissioning process comprises the integrated application of a set of engineering techniques and procedures to check, inspect and test every operational component of the project, from individual functions, such as instruments and equipment, up to complex amalgamations such as modules, subsystems and systems.

Commissioning activities, in the broader sense, are applicable to all phases of the project, from the basic and detailed design, procurement, construction and assembly, until the final handover of the unit to the owner, including sometimes an assisted operation phase.

Measurement and Verification

Measurement and Verification (M&V), sometimes also referred to as monitoring and verification, is a process, which is used to determine energy and demand savings. The primary application of M&V is in those energy efficiency projects where the return on the capital investment is tied to the projected energy savings that will be achieved. M&V becomes a central part of a contract if the contract payments or performance guarantee in a project are dictated by the magnitude of the energy savings that will result from the implementation of a set of energy efficiency measures. M&V is primarily focused on risks that affect the measurement or determination of savings from energy or water efficiency programs.

These risks are defined in the terms of the contracts between the participants. Energy or demand savings are determined by comparing measured energy use or demand before and after implementation of an energy savings program. In general:

Energy Savings = Base year Energy Use — Post-Retrofit Energy Use + Adjustments

The “Adjustments” term in this general equation brings energy use in the two time periods to the same set of conditions. Conditions commonly affecting energy use are weather, occupancy, plant throughput, and equipment operations required by these conditions. Adjustments may be positive or negative.

Adjustments are derived from identifiable physical facts. The adjustments are made either routinely such as for weather changes, or as necessary such as when a second shift is added, occupants are added to the space, or increased usage of electrical equipment in the building. Adjustments are commonly made to restate base year energy use under post-retrofit conditions.

Solved Example:

For the following tasks, durations, and predecessor relationships in the following activity table,

a) Draw the network

b) Calculate expected time for all tasks

c) Calculate variance for all tasks

d) Determine all possible paths and their estimated durations

e) Identify the critical path

e) Duration of critical path is 39 weeks.

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Chapter 7

Chapter 9