Project Management
Week 5: Resource Allocation
2021/1/18
Tran Van Ly
Industrial Engineering & Management
International University
Email: tvly@hcmiu.edu.vn
Room A2-504
1
Recall previous week
Planning tasks
Define
activities
Sequence
activities
Estimate
activities
resources
Estimate
activities
durations
Develop
schedule
2
Recall previous week
Activity
Immediate
Predecessors
A
-
B
-
C
A
D
A
E
A
F
B-C
G
B–C
H
E–F
I
E–F
J
D–H
K
G-I
D
J
A
H
E
Start
C
F
End
I
K
B
G
Recall previous week
SEQUENCE ACTIVITIES
D
J
A
S
H
E
C
F
E
I
K
B
G
a = optimistic estimate
m = most likely time estimate
b = pessimistic time estimate
Activity Immediate Predecessors
A
-B
-C
A
D
A
E
A
F
B,C
G
B,C
H
E,F
I
E,F
J
D,H
K
G,I
Mean (expected time): t=
2
b
a
Variance:  2 =
6
2
a + 4m + b
6
= бA2 + бC2 + бF2 + бI2 +бK2
Recall previous week
Probabilistic Network Analysis
Determine probability that project is
Z
completed within specified time
x-

x
Z
=
where

 = tp = project mean time
 = project standard deviation
x = proposed project time
Z = number of standard deviations x
is from mean
days
Gantt Chart
• A Gantt chart is a horizontal bar chart that
visually represents a project plan over time.
Modern Gantt charts also show the status and
who’s responsible for—each task in the project.
Key Element of Gantt Chart
• Task list: Runs vertically down the left of the
gantt chart to describe project work and may be
organized into groups and subgroups
• Timeline: Runs horizontally across the top of
the gantt chart and shows months, weeks, days,
and years
• Dateline: A vertical line that highlights the
current date on the gantt chart
• Bars: Horizontal markers on the right side of the
gantt chart that represent tasks and show
progress, duration, and start and end dates
Key Element of Gantt Chart
• Milestones: Yellow diamonds that call out
major events, dates, decisions, and deliverables
• Dependencies: Light gray lines that connect
tasks that need to happen in a certain order
• Progress: Shows how far along work is and may
be indicated by % Complete and/or bar shading
• Resource assigned: Indicates the person or
team responsible for completing a task
LINEAR PROGRAMMING APPROACH FOR
CPM ANALYSIS
• Example LP model: MIN
Subject to:
t2 - t1 >= 5;
t4 - t2 >= 8;
t4 - t1 >= 7;
t6 - t5 >= 5;
t3 - t2 >= 0;
t6;
activity A;
activity C;
activity E;
activity G;
dummy activity;
t3 - t1 >= 3;
t4 - t3 >= 7;
t5 - t4 >= 4;
activity B
activity D
activity F
t1 = 0;
2
C, 8
A, 5
1
B, 3
3
E, 7
D, 7
4
F, 4
5
G, 5
6
Using Microsoft Project software Interface
Using Microsoft Project software
Key steps
Using Microsoft Project software
Using Microsoft Project software
Using Microsoft Project software
Using Microsoft Project software
Learning Objectives
• Critical path method –
Crashing a project
• Resource allocation problem
• Resource loading
• Resource leveling
• Constrained resource
scheduling
16
Critical Path Method
Crashing a Project
• CPM includes a way of relating the
project schedule to the level of physical
resources allocated to the project
• This allows the project manager to trade
time for cost, or vice versa
• In CPM, two activity times and two costs
are specified, if appropriate for each
activity
Critical Path Method
Crashing a Project
• The first time/cost combination is called
normal, and the second set is referred to as
crash
• Normal times are “normal” in the same
sense as the ‘m’ time estimate of the three
times used in PERT
• Crash times result from an attempt to
expedite the activity by the application of
additional resources
Critical Path Method
Crashing a Project
• Careful planning is critical when attempting
to expedite (crash) a project
• Expediting tends to create problems; and
the solution to one problem often creates
several more problems that require
solutions
• Some organizations have more than one
level of crashing
© 2006 John Wiley and Sons, Inc.
Chapter 9-3
CRASHING A PROJECT
• Project crashing is a method for shortening the
project duration by reducing the time of one or
more of the critical project activities to less than its
normal activity time.
• The objective of crashing is to reduce project
duration while minimizing the cost of crashing.
• Two simple principles:
– focus on the critical path(s) when trying to
shorten the duration of a project
– when shortening a project’s duration, select the
least expensive way to do it
CRASHING A PROJECT
• Steps in crashing:
1. Compute the crash cost per time period. If
crash costs are linear over time:
Crash cost – Normal cost
Crash cost per period =
Normal time – Crash time
2. Using current activity times to find the
critical path and identify the critical
activities.
CRASHING A PROJECT
• Steps in crashing:
3. Select the activity to crash:
• If there is only one critical path: Select the activity on this
critical path that:
–Can still be crashed
–Has the smallest crash cost per period
4. If there is more than one critical path: Select one activity from
each critical path such that
–Each selected activity can still be crashed
–The total crash cost of all selected activities is the smallest
–Note - the same activity may be common to more than one
critical path
5. Update all activity times. If the desired due date has been
reached, stop. If not, return to Step 2
CRASHING A PROJECT
• The network and durations given below shows the normal
schedule for a project.
Activity
Normal duration Crash duration
(days)
(days)
Normal cost
($)
Crash cost
($)
A
120
100
12000
14000
B
20
15
1800
2800
C
40
30
16000
22000
D
30
20
1400
2000
E
50
40
3600
4800
F
60
45
13500
18000
CRASHING A PROJECT
• The owner wants you to you to finish the project in 110 days.
• Find the minimum possible cost for the project if you want to
finish it on 110 days.
• (Assume that for each activity there is a single linear, continuous
function between the crash duration and normal duration
points).
CRASHING A PROJECT
• Using the normal duration (ND), crash duration (CD), normal
cost (NC), and crash cost (CC), calculate the crash cost slope
for each activity:
Activity
Normal duration
(days)
Crash duration
(days)
Normal
cost ($)
Crash cost
($)
Cost slope
($/day)
A
120
100
12000
14000
100 (3)
B
20
15
1800
2800
200 (4)
C
40
30
16000
22000
600 (6)
D
30
20
1400
2000
60 (1)
E
50
40
3600
4800
120 (2)
F
60
45
13500
18000
300 (5)
CRASHING A PROJECT
• The normal cost for the project is $48300 and the normal
duration is 140 days.
• The first crashing activity is D.
• Maximum of 10 days can be cut from this schedule by reducing
the duration of activity D to the crash duration of 20 days.
CRASHING A PROJECT
• The next crashing activity is E.
• Crashing the activity E by 10 days will cost an additional $120
per day or $1200.
• The project duration is now 120 days and the total project
cost is $50100.
CRASHING A PROJECT
The next stage of crashing requires a more through analysis
• Activity A is paired with each of the other activities to determine
which has the least overall cost slope for those activities which
have remaining days to be crashed
– Activity A ($100) + activity B ($200)
– Activity A ($100) + activity C ($600) + activity F ($300)
• The least-cost slope will be activity A + activity B for a cost
increase of $300 per day. Total project cost would be $51600
CRASHING A PROJECT
• Final step in crashing the project to 110 days would be accomplished
by reducing the duration of activity A by 5 days to 110 days, reducing
activity C by 5 days to 35 days, and reducing activity F by 5 days to 55
days.
• The total project cost for the crashed schedule to 110 days of
duration would be $56600
Crashing – Sample Network
3
8
6
3
2
1
6
10
4
7
0
11
Activity
Normal Time
6
5
Normal Cost
Crash Time
5
Crash Cost
1-2
3
$50
2
$70
2-3
6
$80
4
$160
2-4
10
$60
9
$90
2-5
11
$50
7
$150
3-6
8
$100
6
$160
5-7
5
$40
4
$70
6-7
6
$70
6
$70
Fast-Tracking
• Another way to expedite a
project is known as “fast-tracking”
• It refers to overlapping the design and
build phases of a project
• Because design is usually completed before
construction starts, overlapping the two
activities will result in shortening the project
duration
The Resource Allocation Problem
• A shortcoming of most scheduling procedures is
that they do not address the issues of resource
utilization and availability
• Scheduling procedures tend to focus on time
rather than physical resources
• Time itself is always a critical resource in project
management, one that is unique because it can
neither be inventoried nor renewed
The Resource Allocation Problem
• Schedules should be evaluated not merely in
terms of meeting project milestones, but also
in terms of the timing and use of scarce
resources
• A fundamental measure of the project
manager’s success in project management is
the skill with which the trade-offs among
performance, time, and cost are managed
The Resource Allocation Problem
• The extreme points of the relationship between time use
and resource use are these:
– Time Limited: The project must be finished by a certain time,
using as few resources as possible. But it is time, not resource
usage, that is critical
– Resource Limited: The project must be finished as soon as
possible, but without exceeding some specific level of resource
usage or some general resource constraint
The Resource Allocation Problem
• If all three variables - time, cost, specifications are fixed, the system is “overdetermined”
• In this case, the project manager has lost all
flexibility to perform the trade-offs that are so
necessary to the successful completion of
projects
• A system-constrained task requires a fixed
amount of time and known quantities of
resources
Resource Loading
• Resource loading describes the amounts of
individual resources an existing schedule requires
during specific time periods
• The loads (requirements) of each resource type
are listed as a function of time period
• Resource loading gives a general understanding of
the demands a project or set of projects will make
on a firm’s resources
Resource Loading
• An excellent guide for early, rough project
planning
• Because the project action plan is the source of
information on activity precedencies, durations,
and resources requirements, it is the primary
input for both the project schedule and its budget
• The action plan links the schedule directly to
specific demands for resources
Resource Loading
• The PERT/CPM network technique can be modified
to generate time-phased resource requirements
• The project manager must be aware of the ebbs
and flows of usage for each input resource
throughout the life of the project
• It is the project manager’s responsibility to ensure
that the required resources, in the required
amounts, are available when and where they are
needed
Resource Leveling
• Resource leveling aims to minimize the period-byperiod variations in resource loading by shifting
tasks within their slack allowances
• The purpose is to create a smoother distribution of
resource usage
• Several advantages include:
– Less hands-on management is required
– May be able to use a “just-in-time” inventory policy
Resource Leveling
• When resources are leveled, the associated costs
also tend to be leveled
• The project manager must be aware of the cash
flows associated with the project and of the
means of shifting them in ways that are useful to
the parent firm
• Resource leveling is a procedure that can be used
for almost all projects, whether or not resources
are constrained
Resource Leveling & Smoothing
Constrained Resource Scheduling
• There are two fundamental approaches to constrained
allocation problems:
– Heuristic Methods
– Optimization Models
• Heuristic approaches employ rules of thumb that have
been found to work reasonably well in similar situations
• Optimization approaches seek the best solutions but are
far more limited in their ability to handle complex
situations and large problems
Heuristic Methods
• Heuristic approaches to constrained resource
scheduling problems are in wide, general use for a
number of reasons:
1. They are the only feasible methods of attacking the large,
nonlinear, complex problems that tend to occur in the real
world of project management
2. While the schedules that heuristics generate may not be
optimal, they are usually quite good- certainly good enough
for most purposes
Heuristic Methods
• Most heuristic solution methods start with the
PERT/CPM schedule and analyze resource usage
period by period, resource by resource
• In a period when the available supply of a resource
is exceeded, the heuristic examines the tasks in
that period and allocates the scarce resource to
them sequentially, according to some priority rule
• Technological necessities always take precedence
Heuristic Methods
• Common priority rules:
– As soon as possible
– As late as possible
– Shortest task first
– Most resources first
– Minimum slack first
– Most critical followers
– Most successors
– Arbitrary
Heuristic Methods
• Most priority rules are simple adaptations of the heuristics
used for the traditional “job shop scheduling” problem of
production/operations management
• Most heuristics use a combination of rules: a primary rule, and
a secondary rule to break ties
• As the scheduling heuristic operates, one of two events will
result:
– The routine runs out of activities before it runs out of resources
– The routine runs out of resources before all activities have been
scheduled
Optimizing Methods
• The methods to find an optimal solution to the
constrained resource scheduling problem fall
into two categories:
– Mathematical programming
– Enumeration
• Mathematical programming can be thought of
as liner programming (LP) for the most part
Optimizing Methods
• Linear programming is usually not feasible for
reasonably large projects where there may be a
dozen resources and thousands of activities
• In the late 1960s and early 1970s, limited
enumeration techniques were applied to the
constrained resource problem
• Tree search, and branch and bound methods were
devised to handle up to five resources and 200
activities
Mathematical Programming
• The three most common objectives of mathematical
programming are:
1. Minimum total throughput time (time in the shop) for all projects
2. Minimum total completion time for all projects
3. Minimum total lateness or lateness penalty for all projects
• These objectives are most appropriate for ‘job shop’
type solutions to resource constraints
Heuristic Techniques
• There are scores of different heuristic-based
procedures in existence
• They represent rather simple extensions of wellknown approaches to job-shop scheduling:
–
–
–
–
–
Resource Scheduling Method
Minimum late finish time
Greatest resource demand
Greatest resource utilization
Most possible jobs
Critical Chain
• Eliyahu M. Goldratt’s “Theory of Constraints”
• Traditional Project Estimation Techniques
Ineffective
– Time and Resource Constraints Usually Violated
– PMs Rely on “Padding” of Schedules and Budgets
– Unknown Nature of Event Interaction
• Fear, Uncertainty, Doubt
• Psychological, Organizational, and Physical
Critical Chain - Approach
• Bottleneck Management
– Activities with Several Predecessors and/or
Successors
– Add “Time Buffers” at Bottleneck Events
• “Safety Stock” Equivalent in Manufacturing
• Just-in-Time with “Just-in-Case”
• Statistically-derived “Path Buffers”
– Establish the Critical Chain for Scarce Resources
– Prioritization of Resources in Chain Events
HW1 - CRASHING A PROJECT
• Find the total completion time of the project.
• If we want to complete the project in 10 weeks, which
activities should be crashed?
The activities necessary for the completion of this project are
listed in the following table:
Activity
Immediate
predecessor
Normal time
(weeks)
Crash time
(weeks)
Normal cost
($)
Crash cost
($)
A
-
4
3
2000
2600
B
-
2
1
2200
2800
C
A
3
3
500
500
D
B
8
4
2300
2600
E
B
6
3
900
1200
F
C, D
3
2
3000
4200
G
E, F
4
2
1400
2000
HW2- CRASHING WITH ADDITION BUDGET
The following table provides the information necessary to construct a
project network and project crash data:
Activity
1
2
3
4
5
Activity
Predecessor
1
3
Activity
Time (weeks)
Normal
Crash
20
8
24
20
14
7
10
6
11
5
Activity
Cost ($)
Normal
1000
1200
700
500
550
Crash
1480
1400
1190
820
730
a. Determine the maximum possible crash time for the network, and
crash the network the maximum amount possible.
b. Compute the normal project cost and the cost of the crashed project.
c. Owner has $5000 to speed up the project: Identify which activities to
crash [Develop a mathematical model to optimize crashing cost when
owner ready to pay more than normal cost of $N*1000 (Cost Time
Trade – Off)]