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CPM, Crashing, Resource Leveling

using MS Excel & MS ProjectDavid S.W. LaiSept 24, 2013

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Scope

Linear Programming (LP) approachMS Excel 2010

MS Project 2010

• Critical Path Analysis

• Crashing

• Resource leveling

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Critical Path MethodA Linear Programming

Approach

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Activity Immediate Predecessors

Expected Time (days)

1 Walls and Ceiling 2 52 Foundation - 33 Roof Timbers 1 24 Roof Sheathing 3 35 Electrical Wiring 1 46 Roof Shingles 4 87 Exterior Siding 8 58 Windows 1 29 Paint 6, 7, 10 2

10 Inside Wall Board 8, 5 3

The example description is modified from the exercises described in Moore and Weatherford, Decision Modelling, Pearson 2001.

The Build-Rite Construction Company has identified the following ten activities that take place in building a house.

Determine the critical path and the critical activities.

Example Problem

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Solution

Critical activities: 1, 2, 3, 4, 6, 9

ActivityEarly Start Schedule Late Start Schedule Total

SlacksES EF LS LF1 3 8 3 8 02 0 3 0 3 03 8 10 8 10 04 10 13 10 13 05 8 12 14 18 66 13 21 13 21 07 10 15 16 21 68 8 10 14 16 69 21 23 21 23 0

10 12 15 18 21 6

The project manager should adjust accordingly the budgets and resource allocations to avoid any delay on these activities.

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Critical Path Method• Step 1: Forward pass • Step 2: Backward pass• Step 3: Calculating slacks

Early Start Schedule

Late Start Schedule

Slacks

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Precedence Constraints

Predecessor Successor Duration of the Predecessor

2 1 3

1 3 5

3 4 2

1 5 5

4 6 3

8 7 2

1 8 5

6 9 8

7 9 5

10 9 3

5 10 4

8 10 2

Objective Functionminimize the project duration.

Constraintse.g. activity 6 precedes activity 9

Decision Variablesstart times of the activities

A LP Model for CPM analysis

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Nodes Decision Variables

ArcsPrecedence Constraints

92 1

3

5

8 7

3 5

2

4

2

83

4

10

6

3

5

2

Longest Path

Optimal Solution

AON network & LP Model

Note that an alternative LP model can be derived from the AOA network.Critical activities can then be identified via sensitivity analysis.

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Parameters

Activity Time (days)1 Walls and Ceiling 52 Foundation 33 Roof Timbers 24 Roof Sheathing 35 Electrical Wiring 46 Roof Shingles 87 Exterior Siding 58 Windows 29 Paint 2

10 Inside Wall Board 3Project Start Time 0

Precedence ConstraintsPredecesor Successor

2 1

1 3

3 4

1 5

4 6

8 7

1 8

6 9

7 9

10 9

5 10

8 10

• The start time of the project• The (expected) times of the activities• Precedence Relations of two activities

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A Linear Programming Approach for

Critical Path AnalysisA Spreadsheet Implementation

Activity Time (days) Start time Finish time1 Walls and Ceiling 5 - -2 Foundation 3 - -3 Roof Timbers 2 - -4 Roof Sheathing 3 - -5 Electrical Wiring 4 - -6 Roof Shingles 8 - -7 Exterior Siding 5 - -8 Windows 2 - -9 Paint 2 - -

10 Inside Wall Board 3 - -Project Start Time 0

Objective Value

Start time Finish time3 80 38 10

10 138 12

13 2110 158 10

21 2312 15

23 days

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• Early start schedule

Any activity will be started at its earliest start time.

• Late start schedule

Any activity will be started at its latest start time.

ES EF3 80 38 10

10 138 12

13 2110 158 10

21 2312 15

LS LF3 80 38 10

10 1314 1813 2116 2114 1621 2318 21

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Critical Activities

Early Start/Late Start ScheduleTotal Slacks

Activity ES EF LS LF1 3 8 3 8 02 0 3 0 3 03 8 10 8 10 04 10 13 10 13 05 8 12 14 18 66 13 21 13 21 07 10 15 16 21 68 8 10 14 16 69 21 23 21 23 0

10 12 15 18 21 6

Since the total slacks can be determined using the early start schedule and the late start schedule, the critical activities can be identified as well.

Critical activities: 1, 2, 3, 4, 6, 9

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Demo• To enable the solver in EXCEL 2010– File Options Add-Ins Select “Solver Add-

in” Go Select “Solver Add-in” OK• You may find the solver in– Data Solver Objective

Function

Decision Variables

Constraints

Use simplex method for the LP models

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CrashingA Linear Programming

Approach

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Activity

Normal

Time

Normal

Cost

Crash Time

Crash Cost

1 5 50 3 722 3 20 2 303 2 15 1 304 3 8 1 205 4 30 4 306 8 13 4 217 5 45 1 658 2 45 1 529 2 40 2 40

10 3 22 2 34

2 3 4 5 640

60

80

Activity TimeC

ost

e.g. Cost for Activity 1

Build-Rite’s engineers have calculated the cost of completing each activity. Their results are given below.

How much would it cost to complete the project within 22 days? 21 days? 20 days?...

Example Problem

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Solution: Time-Cost Trade-Off

Project Duration

ProjectCost

13 18 23280

300

320

340

360

380

The normal schedule obtained using CPM

each activity is performed at its lowest cost and at a normal duration.

The crashing process has revealed a relationship between the cost and the schedule of the project, which allows us to prepare our budget by considering the possible trade-offs between cost and time.

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Notations

crashMax. Crash Days

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A LP Model for Crashing with a fixed project due date

Precedence Constraints

Max. Clashed Days

Project due date

Minimize the cost for crashing

No. of days to crash

Start times of the activities.

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CrashingActivity Crashing

(days)Normal

TimeNormal

CostCrash Time

Crash Cost

Max. Crash Days

Cost per Crash Day

1 ? 5 50 3 72 2 112 ? 3 20 2 30 1 103 ? 2 15 1 30 1 154 ? 3 8 1 20 2 65 ? 4 30 4 30 0 06 ? 8 13 4 21 4 27 ? 5 45 1 65 4 58 ? 2 45 1 52 1 79 ? 2 40 2 40 0 0

10 ? 3 22 2 34 1 12Crash Cost ?

Project Cost ?

Obj. Value ?

A Linear Programming Approach for Crashing

A Spreadsheet Implementation

Crashing (days)

2112041001

84288

372.1

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Demo

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Resource LevelingMS Project 2010

22The example question is modified from Project Management (Shtub, Bard, Globerson) Exercise 10.1

Activity Immediate Predecessor

s

Expected Time (days)

Work hours

1 Walls and Ceiling 2 5 30 hrs2 Foundation - 3 22 hrs3 Roof Timbers 1 2 8 hrs4 Roof Sheathing 3 3 16 hrs5 Electrical Wiring 1 4 6 hrs6 Roof Shingles 4 8 4 hrs7 Exterior Siding 8 5 6 hrs8 Windows 1 2 12 hrs9 Paint 6, 7, 10 2 8 hrs

10 Inside Wall Board 8, 5 3 4 hrs

The working hours requirements of the activities are estimated. They are described below.

Example

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The resource profile after leveling.• Minimized resource fluctuation• No delay in the project

Resource leveling

The resource profile before leveling.• large resource fluctuation• Overallocation of resource

The reallocation of slacks in activities to minimize fluctuations in resource requirement profile.

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Demo

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1. Create a Project.

Set the working hours per day. E.g. 8 hours.

The durations of activities (or tasks) are fixed in our case.

1. File New Blank Project

2. File Options Schedule

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2. Input the task information

1. Task Gantt Chart 2. Input the task

information3. Select all the tasks

and then press “Auto Schedule”

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3. Set the Project Start Date• Project Project Information

Statistics

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3. Identify the critical path • Task Gantt Chart Network

Diagram• Gantt chart Add New Column

“total slack” Note that the project can be finished within 23 days.

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4. Add a renewable resource

• Task Gantt Chart Resource Sheet• In the first row, input – Resource Name: Manpower– Type: work–Max. Units: 100%

Examples of renewable resource• Manpower• Materials• Machines

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5. Type in the resource usage

• Input the work hours of the activities • Select the column right click Assign

Resources Select “Manpower” AssignTask Work hour

1 30 hrs2 22 hrs3 8 hrs4 16 hrs5 6 hrs6 4 hrs7 6 hrs8 12 hrs9 8 hrs10 4 hrs

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6. Resource Graph• Task Gantt chart Resource Graph

Large frustration

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7. Resource Leveling• Resource Leveling Options tick

“level only within available slack.• Resource level all

• Frustration is minimized.

• No over-allocation • The project

duration remains the same (total slacks are reduced )

Smaller ffrustration