What Is the Shortest Duration Required to Complete anã¢â‚¬â€¹ Activity

Developed by Selma Lind Jonsdottir

According to the Project Direction Constitute (PMI) , projection management is the application of cognition, skills, tools and techniques to projection activities in social club to meet project requirements and objectives^[i]. The challenging task of managing projects can exist supported by an operation research technique called the Disquisitional Path Method (CPM) . The CPM is a mathematically-network based algorithm in which can be used for planning, scheduling and monitoring projection progress.^{[2] [3]}

The technique developed in late 1950s utilizes data from a work breakdown structure (WBS) in a network representation to brandish interrelationships and dependencies betwixt projection activities that must exist achieved to complete a project. The technique is used for analyzing projects by determining the longest sequence of tasks through a project network, called the critical path. This determines the shortest possible time to complete the project too as which activities should exist given a item focus within the projection.^{[ii] [4]}

Based on acquired information from the CPM, the adjacent question could be if it is possible to shorten the project in order to finish inside certain deadline and what is the least expensive manner to do information technology. In today's competitive environment, there is an increasing pressure to get projects washed earlier and quicker, either from the very showtime, prior to setting the project baseline, or in the midst of project execution in lodge to go the projection back on schedule. Comparison of the benefits by reducing project fourth dimension with the total toll associated with it can often be a huge challenge for project managers, thus, the conclusion to reduce the project elapsing should exist based on an analysis of the trade-offs betwixt time and price.^{[2] [3] [v]}

Contents

• 1 Overview
  • 1.1 Introduction
  • 1.2 Background
• 2 Methodology
  • 2.one Pace 1: Developing the Project Network
  • ii.2 Step 2: Constructing the Project Network
  • ii.three Step 3: The Ciphering Procedure
  • 2.4 Step 4: Critical Path Identification
  • 2.5 Step 5: Update of project progress
• 3 CPM Example
  • 3.1 Constructing the CPM Project Network
  • 3.2 The CPM Computation Process
  • iii.three Slack Computation and Critical Path Identification
• iv Consideration of Fourth dimension-Cost trade-offs
• five Advantages and Benefits of the CPM
• 6 Disadvantages and Limitations of the CPM
• 7 Annotated Bibliography
• 8 References

Overview

Introduction

Organizations beyond the world and within dissimilar sectors have been using projection management as a way to improve project results. The increased awareness and acceptance supports that the awarding of knowledge, skills, tools and techniques can accept meaning impact on projection success.^{[1] [half-dozen]}

Figure i: The Projection Management Iron Triangle ^[7]

Co-ordinate to the ISO 21500 a project is defined as a "unique ready of processes consisting of coordinated and controlled activities with showtime and end dates, performed to achieve projection objectives. Accomplishment of the projection objectives requires the provision of deliverables conforming to specific requirements." ^[8]

Each and every project is carried out nether sure constraints. The 3 constraints of price, fourth dimension and scope represent together the iron triangle of project management, which tin be seen in Figure one. Each constraint forms the vertices with quality as the central theme and together indicates that projects must be delivered inside agreed time and cost and furthermore to meet the agreed scope and client quality requirements. The iii constraints are aligned and changes in ane volition likely impact the others, or bear on the quality of the projection. This emphasizes the importance of project direction and its challenging area in which is not at all easy to deal with.^{[seven] [9]}

Project management is accomplished through the appropriate awarding of five processes: ^[ane]

1. Initiating
2. Planning
3. Executing
4. Monitoring and Controlling
5. Closing

Each and every system tin can be involved in a project of any size, duration and complexity level at any given fourth dimension. It is notable that projection planning, scheduling and monitoring is a major part involved in projection management. Therefore the critical path method (CPM) can be of skilful support inside the challenging procedure of managing projects and helps those involved to address questions such every bit: ^[x]

• What is the (minimum) total time required to complete the whole project?
• What are the (earliest and latest) outset and completion times for private activities in the project?
• Which activities are critical and must be completed on time in club to complete the whole projection on fourth dimension?
• How much delay is tolerated of non-critical activities without impacting the overall project schedule and project completion fourth dimension?
• With cost information on each activeness: What is the least expensive fashion to reduce project duration in gild to meet a targeted completion time?

All these questions are extremely valid and of import to have nether control when managing projects.

Background

The critical path method (CPM) is a mathematically-network based project modeling technique developed in tardily 1950s in order to plan, schedule and control large, circuitous projects with many activities. CPM is ane approach of network techniques that has been widely used and was developed by a DuPont engineer Morgan R. Walker and a Remington Rand reckoner good, James Due east. Kelly, Jr., to manage plant maintenance and construction piece of work. Furthermore, PERT (Programme Evaluation and Review Technique) , a critical path network technique was adult simultaneously and independently from the CPM by the U.S. Navy for managing the Fleet Ballistic Missile (Polaris) submarine project. PERT is commonly used in conjunction with CPM and the ii optimization techniques are often referred to collectively. ^{[3] [11] [12]}

The CPM technique has been used for many different forms of projects including construction of a new building, span or a route, construction of IT-systems, building of ships and research and evolution of new products. As can be imagined, these projects consist of multiple activities that have to exist completed in order to finish the project as a whole. Information technology is important to note that the methodology is normally not applied to real big scale and complex bug with many activities, since developing the project network and applying the technique can be extremely time consuming. Today, there exists project direction software packages based on the methodology, for case Microsoft Project , which are used to calculate all completion times based on a input. The programs are farther able to create a visualization of the whole project which is used to ease overview and understanding, every bit well every bit supporting the monitoring of the project. On the other hand, the methodology can easily be practical manually to smaller problems in which will be the case in this article. Despite what already mentioned, a knowledge of the concept is important, thus the ones who manage projects have to bargain with multiple challenges within planning, coordinating and monitoring all project activities in order to consummate the project of involvement successfully inside agreed time, cost and scope every bit well equally according to customer quality requirements.^[3]

Methodology

The Critical Path Method (CPM) can be divided into the following steps:

• STEP i: Developing the Project Network

• STEP 2: Amalgam the Project Network

• STEP 3: The Computation Process of frontward pass, backward laissez passer and slack

• STEP 4: Critical Path Identification

• STEP 5: Update of Project Progress

The methodology will be explained step-past-pace every bit well as with an case in Chapter 3: CPM Example.

Step one: Developing the Project Network

When developing the projection network, the post-obit input requirements are needed in order to construct a network model of the project of interest: ^[13]

• A listing of all activities required to complete the project likewise known as the work breakdown structure (WBS)
• The duration of each individual activity for completion
• The dependencies betwixt the activities

With this information, the critical path method (CPM) is able to determine the longest path of activities through the project network, defined every bit the critical path and the corresponding activities are defined equally critical activities. Based on the fact that the entire project cannot cease until all the activities are completed, the longest path or the critical path gives besides the minimum required time to complete the unabridged project.

Step 2: Constructing the Project Network

The 2nd step involves cartoon the project network past placing the activities that need to exist completed in the correct logical sequence based on their interdependencies. The CPM note used for each and every activity in the project network can be seen in Figure 2. Therefore, for all activities in the network the following data is registered and/or computed:

Figure two: Activity Notation in a Project Network ^[2]

• ID: Identification code of individual activeness based on the WBS
• Description: Clarification of individual activity based on the WBS
• Elapsing (DUR): Duration time of individual activeness
• Early Start (ES): The earliest starting time of an private action
• Early Finish (EF): The earliest finishing time of an individual action
• Late Outset (LS): The latest starting time of an individual activity
• Late Finish (LF): The latest finishing time of an individual activity
• SL (Slack): The tolerance of delay of an private activity without affecting project completion date

Pace 3: The Computation Procedure

The tertiary pace involves the 2 procedures of forward pass used to compute earliest times, ES and EF, and the backward laissez passer used to compute latest times, LS and LF as well every bit slack computation. ^{[2] [10]}

Frontwards Pass begins at the initial action and traces each path through the network step-by-footstep, to the end of the last project activity/activities. An of import rule of the forward laissez passer procedure is that all immediate predecessors must be completed before a succeeding activity can start. That is, early start (ES) of an activity equals the largest early on finish (EF) of the immediate predecessors.

For the starting action/activities of the network, early start (ES) is set up equally ES = 0 (or another known value) and each path is traced forrad to the finishing activity/activities by applying EF = ES + DUR to all activities in the project network. When finished, the longest path denotes the project completion time for the plan, that is, largest EF = Total Time.

Backward Pass is initiated later on the forrard pass has been completed and starts with the last projection activity/activities without successors and the late finish (LF) is set as equal to the maximum early finish (EF) of those activities. An of import rule of the backward pass is that at the latest time a succeeding activity can start is only when all firsthand predecessors are completed. That is, tardily cease (LF) for an preceding activity equals the smallest late start (LS) of the immediate successor activities.

With that in mind, each path is traced backward towards the initial starting action/activities by applying LS = LF - DUR. When this has been applied to all the activities in the projection network, the total slack of an activeness can exist adamant by using SL = LS – ES or LF – EF. ^[10]

Step 4: Critical Path Identification

The critical path is of keen importance since all critical activities must be completed every bit scheduled in guild to come across the scheduled project completion time, that is, if an action on the critical path is delayed, the project is delayed by the aforementioned amount of time. This is due to the fact that the activities on the disquisitional path accept a zero slack. Slack (SL) is divers as the amount of time an activity tin be delayed without delaying the whole project. Therefore, project managers demand pay close attention to the critical path activities, which can represent around 10% of all projection activities, in society to ensure that they are non delayed and therefore put the unabridged project in the risk of being tardily. ^[2]

Step 5: Update of projection progress

Every bit the project progresses, the actual activity completion times will be known and the project network can be updated in gild to include this information. Information technology is of import to note that a new critical path may emerge and changes may be made in the network if project requirements suddenly modify. ^[14]

The sally of new critical path is related to the sensitivity of a project network. A network is considered sensitive when it has 1 or more disquisitional path(s) and/or non-critical activities with footling slack and therefore it is more probable that the original disquisitional path(s) will change once the project has started. On the other hand, a network is considered insensitive when it has one critical path and non-critical activities with very big slack, and therefore less likely that the critical path volition change once the project has started. ^[2]

CPM Case

In order to proceeds understanding of the critical path method, the following example is provided.

Constructing the CPM Project Network

Table 1 provides the required information of Pace i, that is, nine activities required to complete a projection of interest too as their dependencies and elapsing times. Based on those information, the projects needs can exist translated into a network diagram in which the CPM can exist utilized. The diagram in Stride two graphically shows the project flow and precedence relationships among all project activities and gives the project manager a visual representation of the entire project.

Table i: Projection Data for the CPM

Activity	Elapsing [Weeks]	Predecessor	Description
A	1	-	Amalgam a base of operations
B	five	A	Build and construct wall construction
C	four	A	Create bearings for the roof
D	10	B, C	Raise the roof
E	five	C	Isolate and close the walls
F	7	D, E	Insert windows and doors
G	5	B, D	Ready floors and lay parquet
H	4	E	Set up the outside expanse
I	6	G, F, H	Paint walls and interior

A network model of the project of interest represented in Table 1 can be seen in Figure 3 where all projection activities accept been placed in the right logical sequence based on their interdependencies and using the action annotation presented in Figure two. This is the initial starting point of the critical path method before computation process, Step 3, takes identify.

Figure 3: Network Representsation of the Project Example presented in Table i

The CPM Computation Process

Following the CPM methodology presented before, the first step of the ciphering process is to perform the forward laissez passer. Action A has no predecessors and therefore we assume that the activity tin start as soon every bit the project can start, at time 0. The primeval stop (EF) date for activity A can now be computed every bit 0 + ane = one. Activities B and C have activity A every bit their firsthand predecessor with early start (ES) as one and early cease (EF) as one + 5 = vi and 1 + 4 = 5. Activity D is a merge activeness since it is preceded by both B and C. The early on start (ES) of a merge activity depends on the largest early on finish (EF) of all activities that merge to it, since the activeness cannot start until both the predeceasing activities have finished. Therefore, early on start of activeness D is max(six_(EF-B),5_(EF-C)) = 6. By standing the forrard pass, the early start (ES) and early on finish (EF) for all activities can be computed as shown in Figure 4. When the frontward laissez passer has been completed, it can exist seen that the early finish (EF) for activity I, or the project in total is 29 weeks.

The second step in the CPM methodology is the backward pass starting with the last project activity, which in this instance is I. The late finish (LF) of the finishing node I is gear up equal to its early finish (EF) of 29 weeks and late kickoff (LS) can be computed as 29 - half dozen = 23. The finishing node I is an immediate successor of activities Chiliad, F and H, hence the latest finishing (LF) fourth dimension for all the 3 activities is set to 23 and their late starts (LS) are computed as 23 - five = 18, 23 - 7 = 16 and 23 - 4 = nineteen. Activity E is a burst activity since information technology is a firsthand predecessor for both activities F and H. Hence, the late finish (LF) for activity East is controlled by the smallest late starts (LS) of the two succeeding activities since activity E cannot finish after than the latest get-go of an firsthand successor. Therefore, late end (LF) of activity Due east is min(16_(LS-F),19_(LS-H)) = 16. If the backward pass is continued until the starting node, the late start (LS) and late finish (LF) for all activities tin exist computed equally shown in Figure 4.

Effigy four: Solution of the Project Example presented in Tabular array i based on CPM - Critical Path: A - B - D - F - I

Slack Computation and Critical Path Identification

Later on the completion of frontward and backward passes, it is possible to determine which activities tin can be delayed by computing the slack for private activities in the project network. The slack for activity A is 0 since there is no difference betwixt ES/EF and LS/LF. On the other hand, the slack for activity C is 2 - 1 = 6 - 5 = 1. The slack for individual activities tin can exist computed as shown in Figure 4. After the slack computation, the disquisitional path can hands exist identified as the longest path through the network with disquisitional activities consisting of null slack. The critical path in this instance is: A – B – D – F – I. . That is, if an activity on the path is delayed, the projection is delayed past the same amount of fourth dimension.

Consideration of Time-Cost trade-offs

Figure 5: Project Fourth dimension-Cost Trade-Off Bend. ^[15]

The consideration of time-cost trade-offs for a project is often considered an essential part of a complete Critical Path Method (CPM) assay ^[16]. Based on caused data from the CPM, the next questions could exist if it is possible to shorten the project in order to finish inside certain borderline and what is the least expensive way to practise it. Project crashing is a procedure to place the cost of reducing project duration and so that comparisons can be made with the benefits of getting the project completed earlier. It requires data of the total project cost, which includes both straight and indirect costs. Shortening the project duration will normally increment the direct costs, which can exist assigned straight to projection activities and commonly represents labor, materials and etc. Meanwhile, a reduction in project elapsing indicates a reduction in indirect costs, necessary costs for doing piece of work, which continues for the life of the projection and is not related to individual project activities. Therefore, the fourth dimension-price trade-off analysis can help to identify the optimum cost-time point betwixt increasing toll of individual activities and decreasing overall project costs. The outcome of such assay results in a time-cost merchandise-off curve every bit shown in Figure 5 which furthermore shows that the optimum projection elapsing can exist determined as the project duration that results in the to the lowest degree total projection cost. ^{[17] [18]}

Effigy six: Activity Cost & Time Human relationship.^[10]

Simplified represensation of the relationship between activity duration and its direct costs tin can be seen in Figure vi. Normal time for an activeness represents estimated activity elapsing under normal conditions retrieved from the CPM and normal cost refers to the respective toll of that action, which implies minimum direct cost. Crashing refers to speeding up the duration of an activity and the crash time is the shortest possible fourth dimension an activity can be completed in. The direct toll for completing an activity in its crash time is chosen crash price and crash point represents the maximum fourth dimension an activity can be compressed. The cost-time relationship is assumed to be linear as shown in Figure 6 and for each action a crash cost per period tin be derived as:

That is, the slope gives information almost the cost per unit of time for shortening an private activity and therefore allows comparison of which critical activities to shorten in club to minimize the total direct cost. ^{[2] [10] [17]}

Later having utilized the CPM, the process for projection crashing involves the following steps: ^{[10] [17]}

1. Compute the crash cost per period for all activities in the project network
2. Notice critical path(s) and critical activities in the projection network
3. In the case of simply one disquisitional path, identify a critical activity with the smallest crash toll per period that can still be crashed. Otherwise, identify critical activity/activities from each critical path with the smallest crash cost per period, which tin can nevertheless exist crashed.
4. Reduce the elapsing of the critical activity/activities identified in stride iii by i time period.
5. Reconstruct or adjust the projection network based on changes made in stride 4. Calculate the total costs = directs + indirect costs. Stop the procedure if the desired completion deadline is reached, otherwise return to step three. Continue until the crash indicate has been reached, that is, no further shortening is possible.

An example of projection crashing where the process is utilized and explained stride-past-footstep is bachelor in Affiliate 8.seven: Project Crashing . The procedure can exist used as a continuation of the example that was provided above in Affiliate iii: CPM Example , with a relatively small change. The merely change required is that the duration fourth dimension of activity A is prepare to 0 since it is considered equally a dummy activity.

For additional knowledge and agreement, Video one below provides a detailed step-past-step guidance on the project crashing procedure. Good examples are equally well provided in Chapter eight.four: Shortening Project Duration .

Video one: Example of Project Acceleration / Activity Crashing with detailed step-by-pace guidance.^[xix]

Advantages and Benefits of the CPM

The critical path method (CPM) has been widely used in planning, scheduling and monitoring of projection progress by a diversity of organizations and industries with neat success. The advantages of the CPM are multiple which tin exist used every bit a framework for projection information and insight used by project managers and members concerning projection time, cost and performance. Following is a list of advantages and benefit reasons of why CPM is utilized in organizations today: ^{[two] [eleven] [12] [twenty]}

• The method encourages all project members to identify and graphically represent all various project activities that need to exist accomplished together with their interrelationship and dependencies in a logical manner. This is beneficial in the planning stage since it requires that the projection is thought through in sufficient details in the long range. This minimizes the chances of overlooking necessary project activities and goals.

• The network representation is a graphic visualization of the unabridged project flow, giving consummate overview in which is understood past all members in the project. Furthermore, smaller projection network diagrams can easily be modified or changed when unexpected events occur equally the projection progresses.

• The method provides an estimate of the minimum project elapsing together with scheduling of individual projection activities needed to complete the project of interest in an efficient mode. Furthermore, the method as well identifies the slack or the tolerance of filibuster without affecting the imposed completion time for individual activities.

• The method provides a basis for documentation standard and enhances the communication of projection plans, schedules and time-price performances.

• The method enables the identification of the critical path(southward) and respective critical activities in which special attending should exist on in the projection due to run a risk of filibuster. This is extremely beneficial in the monitoring stage when tracking the project progress. That means regular updates on project status as well as the disquisitional path(s) with the concept of network sensitivity in mind, since the critical paths practise not necessary remain static for the life of the project.

• The benefit of time-price trade off optimization is possible with cost [normal and crash] data on each activity in the project. That is, the CPM can assistance to identify the steps to be taken in lodge to accelerate a project completion or place the shortest possible time or to the lowest degree possible cost that is needed every bit well as the optimum bespeak of time and price.

• The method provides the basis for scheduling labor and equipment also as budgeting the greenbacks flow of the projection.

Disadvantages and Limitations of the CPM

Despite the multiple advantages and benefits of the CPM mentioned earlier, the method possesses of several limitations or disadvantages which can be listed as following: ^{[2] [12] [21] [22]}

• The CPM and the analysis process tin can become extremely complicated as the scope and complication of the projection increases and without a software it might be hard to manage. For big and complex projects, there tin can exist thousands of activities and dependency relationships and the gamble of making a mistake in the network ciphering condign very high.

• The CPM and network diagrams are highly dependent on it and calculator software in which can be of high initial and usage toll for organizations.

• The method does non include labor, equipment scheduling and resource resource allotment, simply provides the basis for it.

• The method requires fourth dimension in order to develop the project network and relies heavily on project managers and members involved in that stage. Poorly divers projection scope and activities can result in ineffective utilise of CPM, which becomes difficult to manage.

• The method operates based on the assumption that the duration time of private activities in the project are known with certainty, which may not always be the case. This is where the network technique variation of CPM, PERT can exist useful which takes more than skeptical view of the time needed to consummate an private activity.

• The method cannot finer handle changes in the project plan during the projection execution. Changes require that the project network is redrawn and the computation process repeated according to new information midway in the project.

• Critical path(south) is/are non always clear and can sometimes exist hard to identify and follow when monitoring the projection progress. Constant review of the network diagram is necessary to identify the shifting and movement of the critical path(s) over fourth dimension.

Annotated Bibliography

Project Management Found. (2008). A Guide to the Project Management Body of Cognition. ^[1]

This is a recognized formal standard for the project direction profession and describes established norms, methods, processes and practices for projection, program and portfolio management.

Larson, E. Due west & Gray, C. F. (2014). Project Management - The Managerial Process. ^[2]

This book provides a holistic, socio-technical view of project management and focuses on the integration of project management into the organization as a whole. The focus is on the essential tools and processes used to manage projects also every bit the human dimension and how they interact to determine the effectiveness and outcomes of projects. Chapter 6 focuses on the evolution of project program and the utilize of project networks. In add-on, the chapter builds on and extends the network technique by introducing dissimilar lag relationships between project activities. Chapter ix focuses on project elapsing reduction and consideration on fourth dimension-cost merchandise-offs. For a detailed example of the project crashing procedure, the reader is recommended to look up pages 316-318.

Chapter on Deterministic Decision Models. (north.d.). Class material in Network Optimization at DTU autumn 2015. Project Scheduling: PERT/CPM. ^[10]

This chapter based on Deterministic Determination Models is extremely good reading material for better understanding and cognition of the two modeling techniques, PERT/CPM, as well as their connexion/differences. Information technology introduces the basics concepts and describes in a useful and comprehensive style the methodologies behind. Detailed case is provided on project crashing in which tin be used as a continuation on the example provided in a higher place ( Chapter iii: CPM Example ) with relatively small-scale changes. Furthermore, based on that instance, the utilise of linear programming in project acceleration and project scheduling is introduced and explained with an case.

Anderson, E. B., & Hales, R. S., (1986). Critical Path Method Applied to Project Planning: Fire Economic science Evaluation Systems (FEES). ^[xi]

The article provides a good introduction of the critical path method as a technique for planning, scheduling and monitoring projects together with its advantages in all the 3 phases. It provides the basic knowledge of the methodology and the different steps that have to be taken. Additionally to what has been described higher up, the construction of time chart is embedded in the CPM method equally 1 step. Furthermore, the article provides a brusk introduction on resource besides as fourth dimension-cost analysis.

Stelth, P. (2009). Project's Analysis through CPM (Critical Path Method). Isles International University. ^[12]

The focus of the study is to understand and evaluate critical paths and critical chains in a project. The study identifies the salient features of the critical path (CPM) and critical chain method (CCM) together with potential problematic areas. Furthermore, it provides the reader with evaluation of the similarities and differences of the two methods, the power to utilize them in conjunction as well every bit the bear upon the two methods tin can have on project management. The written report covers various points under each of those areas and provides the reader with good insight and knowledge by good and detailed discussion. For more detailed and good summation on the advantages and disadvantages of the CPM, the reader is recommended to look upwardly pages 21-25.

Elbeltagi, E. Construction Direction - Affiliate on Project Time-Toll Trade-Offs. ^[17]

This chapter on project time-cost trade-off gives good understanding of the relationship between those two of import constraints in a project. Information technology provides detailed discussion of the basic concepts of projection & action time-price relationships besides as the projection crashing procedure. The reader is highly recommended to await at the many unlike examples provided based on the project crashing procedure to exercise and gain farther agreement.

References

1. ↑ ^{i.0 ane.ane ane.2 1.iii} Projection Management Constitute. (2008). A Guide to the Project Direction Body of Knowledge. 4th Edition. USA. ISBN 9781933890517
2. ↑ ^{ii.00 2.01 ii.02 two.03 2.04 2.05 2.06 2.07 2.08 two.09 2.10} Larson, Due east. Due west & Gray, C. F. (2014). Projection Direction - The Managerial Process. 6th edition. Usa: NY. ISBN 9781259010705
3. ↑ ^{iii.0 3.i 3.2 3.iii} Larsen, J. & Clausen, J., (2009). Course material in Networks and Integer Programming Supplementary at DTU - Notes to Networks and Integer Programming. Retrieved 04. September 2016 from campusnet.dtu.dk
4. ↑ Newbold, R.C. (1998). Project Management in the Fast Lane – Applying the Theory of Constraint. United states of america: FL. ISBN 9781498738064
5. ↑ Sahu, Thou. & Sahu, 1000. (2014). Toll & Time and Also Minimum Projection Elapsing Using Alternative Method International Review of Applied Engineering Enquiry. five(4), pp. 403-412. Retrieved ten. September 2016. Bachelor Online
6. ↑ Project Management Found. (2010). The Value of Project Management. Retrieved 05. September 2016. Available Online Version
7. ↑ ^{7.0 7.i} Wikipedia. The Project direction Triangle. Retrieved 10.08.2016
8. ↑ International Organization for Standardization (2012) ISO 21500 – Guidance on Projection Management.
9. ↑ Heed Tools. The Iron Triangle of Project Direction. Retrieved 10. September 2016. Bachelor Online
10. ↑ ^{10.0 ten.ane 10.ii x.three 10.4 10.v 10.6} Chapter on Deterministic Decision Models. (due north.d.). Course fabric in Network Optimization at DTU fall 2015. Project Scheduling: PERT/CPM. Available online
11. ↑ ^{11.0 11.1 xi.2} Anderson, Due east. B., & Hales, R. S., (1986). Critical Path Method Applied to Project Planning: Burn down Economics Evaluation Systems (FEES). United States Department of Agriculture. USA. Available Online
12. ↑ ^{12.0 12.i 12.2 12.3} Stelth, P. (2009). Project'due south Analysis through CPM (Critical Path Method). Isles International Academy. Retrieved 06. September 2016. Available Online
13. ↑ Santiago, J. & Magallon, D. (2009). The Critical Path Method - (CEE 320 – VDC Seminar). Stanford University. Retrieved 06. September 2016. Bachelor online
14. ↑ SmartSheet. The Ultimate Guide to the Critical Path Method. Retrieved eighteen.08.2016 from SmartSheets online page
15. ↑ Hegazy, T. (1999). Optimization of construction time – Cost merchandise-off assay using genetic algorithms. Figure of Time-Cost Trade-Offs Retrieved 09. September 2016. Bachelor online
16. ↑ Panagiotakopoulos, D. A CPM Time-Cost Computational Algorithm for Arbitrary Activity Price Functions. Department of Civil Engineering, McGill University, Montreal. Retrieved 20.08.2016.
17. ↑ ^{17.0 17.1 17.ii 17.3} Elbeltagi, Due east. Construction Direction - Chapter on Project Time-Toll Trade-Offs. Retrieved 15. September 2016. Available online
18. ↑ CPM Tutor. (2010). Time-Toll Merchandise-Off. Retrieved 15. September 2016 Bachelor online
19. ↑ Youtube video of Project Dispatch / Activeness Crashing - Project Management. Retrieved 26. September 2016 Bachelor online
20. ↑ The Constructor – Ceremonious Engineering science Domicile. Advantages of Critical Path Method (CPM). Retrieved sixteen. September 2016 Bachelor Online
21. ↑ Mind Tools. Critical Path Analysis and PERT Charts. Retrieved 15. September 2016 Available online
22. ↑ Brilliant Hub Project Managmenet. (2013). Advantages and Disadvantages of Critical Path Method. Retrieved 17. September 2016. Available online

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