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**Spreadsheet Modeling & Decision Analysis**

A Practical Introduction to Management Science 4th edition Cliff T. Ragsdale

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Chapter 14 Project Management

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**Introduction to Project Management**

Projects can be simple (planning a company picnic) or complex (planning a space shuttle launch). Successfully completing a project requires: Knowledge of the tasks involved Accurate estimates of time and resources required Knowledge of physical and logical relations between the various tasks Project management techniques Critical Path Method (CPM) Program Evaluation and Review Technique (PERT) Spreadsheets can be used to manage projects, but dedicated project management software is often more effective.

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**An Example: Lightner Construction**

Tom Lightner owns Lightner Construction, a general contracting company specializing in the construction of single-family residences and small office buildings. Tom frequently has numerous construction projects going on at the same time and needs a formal procedure for planning, monitoring, and controlling each project. He is aware of various project scheduling techniques but has never used them. He wants to see how he might apply such techniques to one of the home-building projects he will be undertaking in the near future. The following slide summarizes each of the major activities required for this project.

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**Summary of Activities Time Immediate Required Predecessor**

Activity Description (in days) Activities A Excavate 3 -- B Lay foundation 4 A C Rough plumbing 3 B D Frame 10 B E Finish exterior 8 D F Install HVAC 4 D G Rough electric 6 D H Sheet rock 8 C, E, F, G I Install cabinets 5 H J Paint 5 H K Final plumbing 4 I L Final electric 2 J M Install flooring 4 K, L

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**An Activity-On-Node (AON) Network**

Install Cabinets A B C D E F G H I J K L M Excavate Lay Foundation Rough Plumbing Frame Finish Exterior HVAC Rough Electric Sheet Rock Paint Final Plumbing Final Electric Install Flooring

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**A Comment of Project Networks**

Projects can also be depicted using Activity-On-Arc (AOA) networks. This book uses AON networks (which the author views as superior to AOA). Some software packages use AOA networks, so you should at least be aware that they exist.

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**An Activity-on-Arc (AOA) Network**

1 2 3 4 5 6 7 8 9 10 11 12 13 Excavate Lay Foundation Rough Plumbing Frame Finish Exterior HVAC Rough Electric Sheet Rock Paint Install Cabinets Final Plumbing Final Electric Install Flooring A B C D G F E H I J K L M

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**Start and Finish Points**

AON networks should have unique start and finish points. A B C D E start finish

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CPM: An Overview A Forward Pass through the network determines the earliest times each activity can start and finish. A Backward Pass through the network determines the latest times each activity can start and finish without delaying completion of the project. The longest path through the network is the “critical path”.

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**Information Recorded for Each Node**

ESTi EFTi i ti LSTi LFTi ti = time required to perform activity i ESTi = earliest possible start time for activity i EFTi = earliest possible finish time for activity i LSTi = latest possible start time for activity i LFTi = latest possible finish time for activity i

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The Forward Pass The earliest start time (EST) for the initial activity in a project is “time zero”. The EST of an activity is equal to the latest (or maximum) early finish time of the activities directly preceding it. The EFT of an activity is equal to its EST plus the time required to perform the activity.

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**Results of the Forward Pass**

25 33 8 E 17 J 38 5 I K 42 4 L 40 2 M 46 A 3 F 21 G 23 6 D 7 10 C B Note: ESTH=MAX(EFTC,EFTE,EFTF,EFTG)=25

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The Backward Pass The latest finish time (LFT) for the final activity in a project is equal to its EFT as determined by the forward pass. The LFT for any other activity is equal to the earliest (or minimum) LST of the activities directly following (or succeeding) it. The LST of an activity is equal to its LFT minus the time required to perform the activity.

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**Results of the Backward Pass**

7 10 3 I 33 38 5 K 38 42 4 22 25 33 38 38 42 A 3 B 3 7 4 H 25 33 M 42 46 4 8 3 3 7 25 33 42 46 E 17 25 8 J 33 38 5 L 38 40 2 17 25 D 7 17 10 35 40 40 42 7 17 F 17 21 4 Note: LFTH=MIN(LSTI,LSTJ)=33 LFTD=MIN(LSTE,LSTF ,LSTG)=17 LFTB=MIN(LSTC,LSTD)=7 21 25 G 17 23 6 19 25

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**Slack = LSTi-ESTi or LFTi-EFTi**

The Critical Path C 7 10 3 I 33 38 5 K 38 42 4 22 25 33 38 38 42 Slack=15 Slack=0 Slack=0 A 3 B 3 7 4 H 25 33 42 46 8 M 4 3 3 7 25 33 42 46 E 17 25 8 Slack=0 Slack=0 Slack=0 Slack=0 J 33 38 5 L 38 40 2 17 25 D 7 17 Slack=0 10 35 40 40 42 7 17 Slack=2 Slack=2 Slack=0 F 17 21 4 21 25 Slack=4 Note: Slack = LSTi-ESTi or LFTi-EFTi G 17 23 6 19 25 Slack=2

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**Determining The Critical Path**

Critical activities have zero slack and cannot be delayed without delaying the completion of the project. The slack for non-critical activities represents the amount of time by which the start of these activities can be delayed without delaying the completion of the entire project (assuming that all predecessor activities start at their earliest start times).

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**Project Management Using Spreadsheets**

The early and late start and finish times for project activities can be done in a spreadsheet using array formulas and circular references.

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Array Formulas An array formula can perform multiple calculations using a range of cells and then return either a single result or multiple results. You create array formulas in the same way that you create other formulas, except that you press [Ctrl]+[Shift]+[Enter] to enter the formula.

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**Array Formula Examples-I**

Let’s compare several standard Excel functions with their equivalent array formulas… Excel Function =SUMPRODUCT(E5:E17,F5:F17) Array Formula =SUM(E5:E17*F5:F17)

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**Array Formula Examples-II**

Let’s compare several standard Excel functions with their equivalent array formulas… Excel Function =SUMIF(E5:E17,">10",F5:F17) Array Formula =SUM(IF(E5:E17>10,F5:F17))

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**Array Formula Examples-III**

Let’s compare several standard Excel functions with their equivalent array formulas… Excel Function =COUNTIF(E5:E17,">0") Array Formula =SUM(IF(E5:E17>0,1))

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**Array Formula Examples-IV**

Let’s compare several standard Excel functions with their equivalent array formulas… Excel Function =SUMXMY2(E5:E17,F5:F17) Array Formula =SUM((E5:E17-F5:F17)^2)

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**Array Formula Examples-IV**

Let’s compare several standard Excel functions with their equivalent array formulas… Excel Function =VARP(E5:E17) Array Formula =AVERAGE((E5:E17-AVERAGE(E5:E17))^2)

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Circular References The array formulas used to implement the project management calculations create circular references. A circular reference occurs when the value in a cell depends on the value in another cell that, in turn, is dependent on the value in the original cell. Usually, a circular reference indicates a formula contains an error – and Excel displays a warning telling you so! Occasionally, a circular reference is needed to accomplish a particular task. This is such an occasion. To tell Excel you intend to use circular references: Click Tools, Options Click the Calculation tab. Select the Iteration option. Click OK.

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**Project Management Example**

See file Fig14-11.xls Learning Tip! Use the Tools, Formula Auditing, Evaluate Formula command to step through the evaluation of the array formulas for the EST and LFT calculations.

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**Important Implementation Issue**

It is important to use activity labels that are unique and do not appear as substrings within other activity labels. For example, the 26 letters of the English alphabet may be used to uniquely identify up to 26 activities in a project. Using the strings "A1" and "A11" as activity labels won’t work The FIND( ) function would locate "A1" within "A11" (i.e., FIND("A1","A11")=1) -- erroneously identifying activity A11 as a predecessor or successor of activity A1. Using the strings "A01" and "A11" as activity labels easily remedies this situation.

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**Important Implementation Issue**

If you use numbers rather than letters to identify activities, using the numbers 1, 2, 3, …, 9 as activity labels would create matching problems within activity labels 11, 12, 13, …, 19 (among others). This can be avoided easily by applying Excel's "Text" format to cells containing activity labels and immediate predecessors and using two-digit numbers for all activity labels (e.g., 01, 02, 03, …, 09, 10, 11, 12, 13, …, 99). If more than 100 numeric activity labels are needed, three-digit numbers (formatted as text) should be used.

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**A Gantt Chart for the Example Problem**

5 10 15 20 25 30 35 40 45 50 A B C D E F G H I J K L M Activity Time Period Activity Time Slack

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Project Crashing It is often possible to complete activities more quickly than normal by applying more resources (better equipment, overtime, etc). This is referred to as “crashing” the project. We may want to determine the optimal way of crashing a project to: complete it more quickly than originally scheduled keep it on schedule if critical activities were delayed

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**Computing Crash Times and Costs**

See file Fig14-14.xls

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**Determining the Earliest Crash Completion Time**

We can determine the earliest possible (crashed) completion time of a project by solving an LP problem…

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**Defining The Decision Variables**

Ti = the time at which activity i begins ti = the normal activity time of activity i Ci = the amount of time by which activity i is crashed

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**Defining The Objective**

Minimize the completion time of the last activity (activity M): MIN: TM + tM - CM

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**Defining The Constraints**

For each arc in the project network from activity i to activity j, we need a constraint of the form: Tj >= Ti + ti - Ci

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**Summary of the Earliest Completion Time Model**

MIN: TM + tm - CM S.T.: TB - TA >= tA - CA TC - TB >= tB - CB TD - TB >= tB - CB TE - TD >= tD - CD TF - TD >= tD - CD TG - TD >= tD - CD TH - TC >= tC - CC TH - TE >= tE – CE TH - TF >= tF - CF TH - TG >= tG - CG TI - TH >= tH - CH TJ - TH >= tH - CH TK - TI >= tI - CI TL - TJ >= tJ - CJ TM - TK >= tK - CK TM - TL >= tK – CL Ti, Ci >= 0, for all i Note: This model can easily be modified to minimize crash costs. Ci <= allowable crash days for activity i

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**Implementing the Model**

See file Fig14-14.xls

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**Cost/Time Trade-Off Curve**

$0 $2,000 $4,000 $6,000 $8,000 $10,000 $12,000 $14,000 $16,000 $18,000 $20,000 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 Completion Time (in days) Crash Cost

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PERT: An Overview CPM assumes all activity times are known with certainty or can be estimated accurately. PERT accounts for uncertainty in activity times by using three time estimates: ai = duration of activity i assuming the most favorable conditions bi = duration of activity i assuming the least favorable conditions mi = estimate of the most likely duration of activity i PERT then estimates expected duration ti and variance vi of each activity’s duration as:

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**PERT Overview Continued**

The expected (or mean) time required to complete any path in the network is the sum of the expected times (the ti) of the activities on the path. Assuming the individual activity times in a project are independent of one another, we may also calculate the variance of the completion time for any path as the sum of the variances (the vi) of the activities on the path. PERT considers the path with the largest expected completion time to be the critical path. PERT’s reasoning may be flawed...

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**PERT Example B A D C a=8 t =9 m=9 v=0.111 b=10 a=2 a=3 t =4 m=4 t =5**

Path: Expected Time: Variance: A - B - D = 18 1.000 A - C - D = 17 4.889

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**Distribution of Completion Times**

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Path Completion Time Probability Density "Critical" Path A-B-D "Non-Critical" Path A-C-D If we want to finish within 21 days, which path is most critical?

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**Simulating a Project Network**

The solution to the “problem” with PERT is to use simulation. We can model activity times easily using a triangular distribution... 1 2 3 4 5 6 Time Required Probability Density a m b

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**Simulating The Project Network**

See file Fig14-25.xls

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Microsoft Project Dedicated project management software such as MS Project can greatly simplify the process of organizing, planning, and controlling projects. A trial version of MS Project is included on the CD-ROM accompanying this book.

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End of Chapter 14

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