1 chapter 14 a construction superintendent explained to his client the massive cost overruns. he...

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Chapter 14

A construction superintendent explained to his client the massive cost overruns. He told of concrete trucks arriving too soon and having to dump their loads; of having to re-pour the foundation because the gravel underlay had not been set; of having to cut many holes through the concrete floor pad because the pipes had been covered before being inspected. Everything happened as scheduled, the poor man lamented. “You should have used PERT,” the client admonished.

Project Planning

with PERT and CPM

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The Importance of Timein Planning

Projects may have uncertainty. Projects may take a long time. Activities must be scheduled.

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Basic Concepts of PERT

PERT has an activity orientation. An activity is an effort that consumes time and resources.

The central focus of all analysis and procedures is the PERT Network.

PERT ties together all project activities as predecessors and successors.

Events are project milestones and serve as the logical glue (network nodes) for connecting activities (network arcs).

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Project Activitiesin Constructing a House

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PERT Network forConstructing a House

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Earliest Possible Event Times (TE)

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Finding TE Values

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Latest Allowable Event Times (TL)

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Finding TL Values

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Critical Pathsand Event Slack

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Activity Scheduling

Project activities need two scheduled times: the starting time and the finishing time.

PERT finds early and late values for each. All are relative to time 0 for the project start.

An activity’s early starting time equals the TE of its beginning event: ES = TE.

An activity’s late finishing time equals the TL for its ending event: LF = TL.

From those, 2 times are computed using the expected activity completion time t:

Early finishing time: EF = ES + t.Late starting time: LS = LF t.

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Scheduling Parameters for Constructing a House

The following apply.

Scheduled times must fall between ES and LF and allow at least time t to complete.

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Activity Slack

The activity slack expresses the scheduling leeway for the activity. It is computed by:

activity slack = LS ES = LF EF Critical activities have zero slack. These lie

on a critical path. Event slack and activity slack measure

different things and are only loosely related.

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Using QuickQuant with PERT

QuickQuant will construct a PERT network and evaluate it using only tabular data input.

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Time Cost Tradeoffs It is possible to speed up a project by crashing some

activities at extra cost. The following data apply.

To reduce project time, speed up one or more activities on the critical path.

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Time-Cost Tradeoffs By successively reducing project time by 1 day at lowest

cost, these plans were found.

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PERT Network for Plan 4

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PERT Network for Plan 7

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Probabilistic Analysis

The activity completion times are really random variables and probabilities apply.

They are well represented by the modified beta distribution. There are three time parameters:

most likely m optimistic a pessimistic b The expected value or mean and variance are

computed from:

t =a + 4m + b

2 =(b a)2

6 62

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Single Path Probabilities The normal distribution represents probabilities

for the duration T of a single path.

Using the above, the mean and variance are computed. From them, probabilities may be found that T lies in any specified interval.

Probabilities for the duration of the a priori critical path can be computed.

WARNING: That will not represent the duration of the project itself. The computed above understates expected project duration.

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Probabilities for Time to Construct a House

The following data apply.

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Probabilities for Time to Construct a House

The expected duration of the critical path a-b-d-e-i-k-l is:

= 5 + 2 + 12 + 10 + 1 + 3 + 9 = 42 days The variance for duration of that path is

2 = .444+.444+7.111+1.000+0+.250+.444 = 9.693and the standard deviation is the square root of the above = 3.11 days

Using 40 days, z = (40 – 42)/3.11 = .64 and Pr[T > 40] = .5 + .2389 = .7389This applies to the critical path, not to the project itself.

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Single-Path Probabilities and QuickQuant

QuickQuant provides a detailed probabilistic analysis of the critical path.

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What’s Wrong withSingle-Path Probabilities?

Single path probabilities do not properly apply to the project itself because many paths might end up to be the longest. The critical path is one of many possibilities. Durations of all these paths are not independent

random variables. To properly assess likelihoods for project

duration, build the house (on paper) many times and see what happens.

QuickQuant will do that. The procedure is called Monte Carlo simulation.

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Simulating House Construction with PERT

QuickQuant provided these simulation results.

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Simulating House Construction with PERT

The second QuickQuant screen tells us about unexpected longest paths.

The a priori critical path was of longest duration only 437 times out of 500. In some projects, it may be longest as little as 1% of the time, or less.

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PERT/CPM Evaluation for Home Construction Example (Figure 14-26)

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A B C D E F G H I J K L M N O P Q R

PROBLEM: Home Construction Example

OnCritical

Activities Name 1 2 3 4 5 6 7 8 9 10 11 Duration Activities Name TEend - TEbeg Path?a Excavating -1 1 5 a Excavating 5 Yesb Pour Foundation -1 1 2 b Pour Foundation 2 Yesc Outside plumbing -1 1 6 c Outside plumbing 24 Nod Framing -1 1 12 d Framing 12 Yese Inside plumbing -1 1 10 e Inside plumbing 10 Yesf Wiring -1 1 9 f Wiring 11 Nog Roofing -1 1 5 g Roofing 5 Noh Brickwork -1 1 9 h Brickwork 17 Noi Plumbing inspection -1 1 1 i Plumbing inspection 1 Yesj Shingling -1 1 2 j Shingling 6 Nok Cover walls -1 1 3 k Cover walls 3 Yesl Interior finish -1 1 9 l Interior finish 9 Yes

m Exterior finish -1 1 7 m Exterior finish 7 Non Landscaping -1 1 8 n Landscaping 11 No

D1 Dummy 1 -1 1 0 D1 Dummy 1 0 No

Project1 2 3 4 5 6 7 8 9 10 11 Time0 5 7 19 29 24 24 30 31 33 42 42 Sum of Earliest Event Times = 244

PERT/CPM EVALUATION RESULTS

Activity - Event TableEvents

Solution (Earliest Event Times, TE j)

89

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O P Q R=A8 =IF(ISBLANK(B8),"",B8) =SUMPRODUCT($C$26:$M$26,C8:M8) =IF(R38=1,"Yes","No")=A9 =IF(ISBLANK(B9),"",B9) =SUMPRODUCT($C$26:$M$26,C9:M9) =IF(R39=1,"Yes","No")=A10 =IF(ISBLANK(B10),"",B10) =SUMPRODUCT($C$26:$M$26,C10:M10) =IF(R40=1,"Yes","No")

26N

=M26

26R

=SUM(C26:M26)

The length of the critical path is in cell N26. It is 42 in this example.

The length of the critical path is in cell N26. It is 42 in this example.

The activities on the critical path are given in column R (indicated by the word Yes). In this case they are a, b, d, e, I, k, and l.

The activities on the critical path are given in column R (indicated by the word Yes). In this case they are a, b, d, e, I, k, and l.

The earliest event times are given in cells C25:M26

The earliest event times are given in cells C25:M26

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OnCritical








89

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26N

=M26

26R

=SUM(C26:M26)

1. Enter the problem name in cell B3.

1. Enter the problem name in cell B3.

2. Expand the table to accommodate the proper number of activities and events by inserting or deleting rows and columns. If rows are inserted, make sure to copy the formulas in columns O, P, Q, and R into these new rows.

2. Expand the table to accommodate the proper number of activities and events by inserting or deleting rows and columns. If rows are inserted, make sure to copy the formulas in columns O, P, Q, and R into these new rows.

3. Enter data in the activity-event table. Put a -1 in the column where the activity starts and a +1 where the activity ends. All other entries in the row are blank. For example, cell C8 has a -1 because activity A starts with event 1 and cell D8 has a +1 because activity A ends with event 2.

3. Enter data in the activity-event table. Put a -1 in the column where the activity starts and a +1 where the activity ends. All other entries in the row are blank. For example, cell C8 has a -1 because activity A starts with event 1 and cell D8 has a +1 because activity A ends with event 2.

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PERT/CPM Evaluation for Home Construction Example (Figure 14-26 )

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OnCritical








89

10


26N

=M26

26R

=SUM(C26:M26)

4. Enter the activity durations in column N (cells N8:N22 in this example).

4. Enter the activity durations in column N (cells N8:N22 in this example).

5. Click on Tools and then Solver to get the Solver Parameters dialog box shown next. 5. Click on Tools and then Solver to get the Solver Parameters dialog box shown next.

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Solver Paramters Dialog Box(Figure 14-27)

1. Enter the project time, cell R26, in the Target Cell line, either with or without the $ signs.

1. Enter the project time, cell R26, in the Target Cell line, either with or without the $ signs.

2. The Target Cell is to be minimized so click on Min in the Equal To line.

2. The Target Cell is to be minimized so click on Min in the Equal To line.

3. Enter the decision variables in the By Changing Cells line, C26:M26.

3. Enter the decision variables in the By Changing Cells line, C26:M26.

4. The constraints are entered in the Subject to Constraints box by using the Add Constraints dialog box shown next (obtained by clicking on the Add button). If a constraint needs to be changed, click on the Change button. The Change and Add Constraint dialog box function in the same manner.

4. The constraints are entered in the Subject to Constraints box by using the Add Constraints dialog box shown next (obtained by clicking on the Add button). If a constraint needs to be changed, click on the Change button. The Change and Add Constraint dialog box function in the same manner.

NOTE: Normally all these entries appear in the Solver Parameter dialog box so you only need to click on the Solve button. However, you should always check to make sure the entries are correct for the problem you are solving.

NOTE: Normally all these entries appear in the Solver Parameter dialog box so you only need to click on the Solve button. However, you should always check to make sure the entries are correct for the problem you are solving.

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The Add Constraints Dialog Box1. Enter Q8:Q22 (or $Q$8::$Q$22) in the Cell Reference line. These are the differences between the earliest times at the ending and beginning nodes for each activity.

1. Enter Q8:Q22 (or $Q$8::$Q$22) in the Cell Reference line. These are the differences between the earliest times at the ending and beginning nodes for each activity.

2. Enter >= as the sign because the differences in ending and beginning earliest times cannot be less than the activity durations, given next in Step 3.

2. Enter >= as the sign because the differences in ending and beginning earliest times cannot be less than the activity durations, given next in Step 3.

3. Enter the activity durations N8:N22 in the Constraint line (or =$N$8:$N$22).

3. Enter the activity durations N8:N22 in the Constraint line (or =$N$8:$N$22).

4. Click the OK button.

4. Click the OK button.

Normally, all these entries already appear. You will need to use this dialog box only if you need to add a constraint.

Normally, all these entries already appear. You will need to use this dialog box only if you need to add a constraint.

CEMCO:CEMCO:

If you need to change a constraint, the Change Constraint dialog box functions just like this one.

If you need to change a constraint, the Change Constraint dialog box functions just like this one.

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OnCritical








89

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26N

=M26

26R

=SUM(C26:M26)

6. To find which activities are on the critical path (in column R) copy the portion of the Sensitivity Report titled "Constraints" and paste it into this spreadsheet so that the word Constraints is in cell N35 (for this example), as shown next.

6. To find which activities are on the critical path (in column R) copy the portion of the Sensitivity Report titled "Constraints" and paste it into this spreadsheet so that the word Constraints is in cell N35 (for this example), as shown next.

7. If all the earliest event times are desired, run Solver once more using the sum of the earliest event times (cell R26) as the target cell.

7. If all the earliest event times are desired, run Solver once more using the sum of the earliest event times (cell R26) as the target cell.

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Solver’s Sensitivity Report

To get Solver’s Sensitivity Report, highlight Sensitivity Report in the Report box of the Solver Results dialog box before clicking the OK button.

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Portion of Sensitivity Report Showing Critical Path (Figure 14-28)

353637383940414243444546474849505152

N O P Q R S T UConstraints

Final Shadow Constraint Allowable AllowableCell Name Value Price R.H. Side Increase Decrease

$Q$8 Excavating TEend - TEbeg 5 1 5 1E+30 5$Q$9 Pour Foundation TEend - TEbeg 2 1 2 1E+30 7$Q$10 Outside plumbing TEend - TEbeg 24 0 6 18 1E+30$Q$11 Framing TEend - TEbeg 12 1 12 1E+30 11$Q$12 Inside plumbing TEend - TEbeg 10 1 10 1E+30 2$Q$13 Wiring TEend - TEbeg 11 0 9 2 1E+30$Q$14 Roofing TEend - TEbeg 8 0 5 3 1E+30$Q$15 Brickwork TEend - TEbeg 20 0 9 11 1E+30$Q$16 Plumbing inspection TEend - TEbeg 1 1 1 1E+30 2$Q$17 Shingling TEend - TEbeg 3 0 2 1 1E+30$Q$18 Cover walls TEend - TEbeg 3 1 3 1 3$Q$19 Interior finish TEend - TEbeg 9 1 9 1 3$Q$20 Exterior finish TEend - TEbeg 7 0 7 3 1$Q$21 Landscaping TEend - TEbeg 8 0 8 3 1$Q$22 Dummy 1 TEend - TEbeg 0 0 0 3 1

Column R shows the critical path: a 1 indicates the corresponding activity is on the critical path and a 0 that it is not.

Column R shows the critical path: a 1 indicates the corresponding activity is on the critical path and a 0 that it is not.

1 chapter 14 a construction superintendent explained to his client the massive cost overruns. he...

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