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Paper ID #33152
An Innovative Approach to Teaching Project Resource Leveling
Dr. David S. Greenburg, The Citadel
Dr. Greenburg is an Associate Professor in the Department of Engineering Leadership and Program Man-agement (ELPM) in the School of Engineering (SOE) at The Citadel. He served over 20 years of activemilitary service in the United States Marine Corps. During his military career he served in a variety of pro-gressively responsible command and staff and leadership positions in Infantry, Logistics, Acquisition, andHuman Resources; with peacetime and combat experience. Upon completion of active military service,Dr. Greenburg served in technical program management and leadership positions at Eagan McAllisterAssociates, and Science Applications International Corporation until he joined the faculty at the Citadel.Dr. Greenburg’s research interests include modeling project networks, technical decision making andleadership. Dr. Greenburg earned is bachelors degree from The Citadel (1981), Masters of Science degreefrom the Naval Postgraduate School (1994), and his PhD in Business Administration (Management ofEngineering and Technology) from Northcentral University (2010). He is a certified Project ManagementProfessional (PMP) by The Project Management Institute (PMI).
Dr. Nahid Vesali P.E., The Citadel
Dr. Nahid Vesali is an Assistant Professor in the Department of Engineering Leadership and ProgramManagement (ELPM) in the School of Engineering (SOE) at The Citadel. She joined the program in Aug2020. She teaches project management, technical planning and scheduling, and construction managementcourses at The Citadel. Dr. Vesali earned her PhD in Civil Engineering from Florida International Uni-versity. She holds M.Sc. in Construction Engineering and Management from IAU, and B.Sc. in CivilEngineering from Iran University of Science and Technology. Prior to joining The Citadel, she worked atPlaza Construction, Florida Group LLC. She worked with the corporate Quality Management team andproject management team for high-rise projects. Dr. Vesali’s past research has been focused on decisionmaking and risk management in existence of deep uncertainty. She is also interested in research relatedto creating inclusive environment for female and minority students in STEM majors.
Dr. Dimitra Michalaka P.E., The Citadel
Dimitra Michalaka, Ph.D., P.E., is an associate professor in the civil and environmental engineering (CE)department at The Citadel, the Military College of South Carolina (SC), U.S.A, the Associate Director forthe Center for Connected Multimodal Mobility (C2M2), and a register professional engineer at the state ofSC. She received her undergraduate diploma in civil engineering from the National Technical Universityof Athens in 2007. Shortly thereafter she moved to the United States to pursue graduate studies at theUniversity of Florida. She graduated with a M.S. in CE in 2009 and a Ph.D. in 2012. Dr. Michalakais passionate about teaching in college and K-12 levels and conducting research in both transportationengineering, focused on traffic operations, congestion pricing, and traffic simulation, and engineeringeducation. Recently, in December 2020, she is also graduated with a Master’s of Science degree inProject Management from The Citadel.
c©American Society for Engineering Education, 2021
An Innovative Approach to Teaching Project Resource Leveling
Abstract
This paper presents an innovative approach using an integrated example for teaching students
effective schedule creation and resource management using Linear Programming. This approach
is designed to deepen student understanding of the various factors that should be considered for
effective project decision making. Successful schedule control requires an understanding of
important project elements to include requirements, schedule, budget, assumptions, constraints,
and objectives. These elements can compete against each other and makes resource allocation
difficult to address without using optimization techniques. Engineering students in introductory
project management courses are typically taught how to identify and sequence activities to
develop a project schedule using simple example problems. These examples tend to be composed
of only a few activities and resources which support development of an initial schedule and often
do not address the impact of scope changes that require reallocation of resources and revisions to
the schedule. Successfully adjusting resource allocation to address project scope and activity
duration changes requires a solid understanding of project crashing. Project crashing involves
shortening project duration by reducing the time of one or more activities. Crashing is done by
increasing the resources to the project, and involves cost and schedule trade-offs. The integrated
example focuses on minimizing the total cost that results from (a) resource allocation, (b)
activity dependencies, (c) timing of deliverables using the available spreadsheets. The method
has been introduced in both undergraduate and graduate introductory project management
classes. Student feedback shows that the method supports concept mastery by demonstrating
how to achieve balanced solutions.
Key Words: Resource Allocation, Scheduling, Optimization, Linear Programming
Introduction
Translating customer requirements into work activities and then developing an effective plan for
accomplishing them is a key aspect of project management. Project managers are challenged
with taking a set of activities that need to be performed in a certain sequence and completed
within a specified time and cost [1]. An effective project schedule provides the framework for
the assignment of resources and sequencing of activities to be performed to carry out the plan of
work [2]. The ability to create resource estimates and build project schedules is an essential
multi-disciplinary management tool and the importance of engineering practitioners and
managers having a working knowledge continues to rise [3]. As the project progresses project
managers and their customers tend to learn additional information that often leads to scope
changes involving revising activity completion times by applying additional resources. The
complexity of schedule control and resource allocation grows significantly as changes to scope
and activity durations occur. Network based techniques such as the Critical Path Method (CPM)
and Project Evaluation and Review Technique (PERT) have proven useful in planning,
scheduling and controlling projects. Reducing project duration often involves adding more
resources to the performance of the activity to shorten its duration while increasing the overall
cost of the project [1].
Schedule development, resource management, and control is taught as part of The Citadel’s
undergraduate and graduate technical project management courses which explore the principles
and applications of project scope, project requirements and work breakdown structures (WBS);
development and control of schedules, risk management, project execution and closeout. The
students are introduced to scheduling as a management tool for recognizing the sequence of
project activities, identifying critical activities, planning efficient use of resources and
completion of the work scope to manage risk. They develop an understanding of how accurate
scheduling reduces variances and downstream problems and contributes to effective project
management. A common cause of variances between schedule estimates and actual work
performance arises when the severity of problem impacts were not seen or known [3]. Students
are also exposed to the use of assumptions and constraints to cover gaps in information that can
cause variance between planned and executed activities.
The specific course learning outcomes include:
1. Developing:
a. Schedule Management Plan
b. Activity List with Attributes and Milestone List
c. Project Schedule Network Diagrams
d. Activity Resource Requirements and Resource Breakdown Structure
e. Project Schedule with Model Data and Schedule Baseline
f. Activity Cost Estimates with Activity Cost Supporting Detail
2. Creating, updating, and analyzing project schedules to determine integrity and validity.
3. Applying resource management techniques to identify resource requirements and to level
resources in order to meet project objectives.
However, test results and end of course feedback reflected that while the students were learning
the mechanics of identifying requirements, creating a project schedule and assigning resources,
they did not necessarily have a full grasp of dealing with the impact of scope changes on the
schedule and resource allocation. In response the faculty brainstormed on ideas they could
incorporate into the instruction to better teach students to create schedules and how to plan for
and manage scope changes. An instructional goal was to develop an integrated exercise that
could be used to demonstrate and tie together key concepts through a comprehensive example
project. The authors decided to develop an integrated spreadsheet model which are easy to use
and provide a natural interface for model building [4].
Description
This paper discusses the development of an integrated exercise to teach students how to create
and control technical project schedules and optimize resource allocation using PERT/CPM and
Linear Programming (LP) techniques. The instruction is designed to complement several existing
courses (PMGT 650, PMGT 651, CIVL 411, and CONE 410).
Due to their ease of use and availability we chose to build a spreadsheet based exercise.
Spreadsheet applications have an advantage over other software programs in that most students
already have spreadsheet programs on their computers. Spreadsheets are also an attractive
technology because students are likely to use spreadsheets in future projects, careers and in
personal life. Spreadsheet programs are flexible, familiar, and relatively easy to use [5]. The
specific PERT/CPM and LP techniques we will discuss in this paper are implemented using the
standard Excel Solver function. We introduce students to the concepts of integrated schedule and
resource optimization using a simple project involving the relocation of a work facility. The
Facility Relocation Project shown in Table 1 consists of ten activities.
Table 1. Facility Relocation Activities and Precedence’s.
Planned Project Start Date 1/2/2020 P1 P2 P3 P4 Max
Activity Task
Working
Days Predecessors Start date End date
S Start 1 1/2/2020 1/3/2020
A Locate Facility 8 S 1/2/2020 1/14/2020
B Order Furniture 9 A 1/15/2020 1/29/2020
C Interview 10 S 1/2/2020 1/16/2020
D Hire & Train 12 C 1/17/2020 2/5/2020
E Remodel 8 A 1/15/2020 1/28/2020
F Furniture Setup 8 B 1/30/2020 2/11/2020
G Inspect 8 D E F 2/12/2020 2/25/2020
H Occupy 8 G 2/26/2020 3/9/2020
I Start Operations 1 H 3/10/2020 3/11/2020
At the start of the exercise students were provided the work scope statement with a project
completion date. They were tasked with identifying the activities, determining precedence
relationships, estimating task duration and developing a project network diagram similar to the
one shown in Figure 1.
Figure 1. Network Diagram for the Facility Relocation Activities.
Students use the Estimate tab in the exercise spreadsheet to calculate a three-point PERT
estimate for each of the activity durations. The spreadsheet uses built in Excel functions to
calculate a 95% confidence interval for the project completion date.
A
B
D
F
E
C G H I J
Figure 2. PERT Estimate for the Facility Relocation Activities.
Once the activity durations and relationships are established and the network diagram is created
students are shown how to calculate the critical path using an LP formulation to determine the
earliest start time (ES) subject to activity precedence. The LP formulation is:
Objective Function: Minimize EST = X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10
Subject to Constraints:
X1 = 0 X2 - X1 >= 1 X3 - X2 >= 8 X4 - X1 >= 1
X5 - X4 >= 10 X6 - X2 >= 8 X7 - X3 >= 9 X8 - X7 >= 8
X9 - X8 >= 8 X10 - X9 >= 8 X1....X10 >=0
Figure 3. LP Formulation to Minimize Early Start Times in Excel.
The Excel Solver formulation for the ED times is shown in Figure 4.
Figure 4. LP Formulation to Minimize Early Start Times in Excel Solver.
Using the ES times students are shown how to calculate the earliest finish (EF) which is the
activity ES plus its duration, the (LF) which is the previous activity’s EF minus its duration, and
the latest start (LS) which is the activity’s LF minus its duration as shown in Table 2. Activity
float is calculated by subtracting ES from LS and EF from LF. Conditional formatting is applied
to the cells to identify those activities on the critical path in red font.
Table 2. LP Formulation to Minimize Early Start Times in Excel.
The result shows that the expected completion time for the project is 43 days (which takes into
account non-workday holidays) and a critical path (CP) of A > B > C > G > H > I. The PERT
calculations reveal that based on the critical path activities there is a 95% confidence interval of
project completion between 41.02 and 45.64 days as shown in Figure 2.
Resource estimates are integrated into the problem by having the students assign various project
personnel to work on each activity. Hourly rates for each personnel resource are multiplied by
the time durations personnel are assigned to work on tasks which feed into the project cost
estimate shown in Table 3. At this point students have created a baseline schedule, identified the
critical path and cost estimate to complete the project. This is a good point to introduce students
to scope changes and the impact of changing client requirements on schedule and costs. Project
managers often have to shorten the planned schedule to expedite the completion date [6]. It is
often valuable to be able to present to the customer options for shortening the project as well as
the resulting cost impacts [7]. We show the students how to accomplish this by expanding on the
initial project calculations and continuing with our Facility Relocation exercise by employing the
crashing technique to shorten the critical path duration. Through the exercise students are tasked
with determining the optimal way of expediting the project by applying additional resources to
shorten activity durations times and evaluating time-cost trade-offs by employing crashing
techniques.
Table 3. Project Resource Allocation and Costs.
In the exercise the cost of crashing specific activities is calculated using a factor of 1.85 times the
normal activity cost multiplied by the change in duration due to crashing as shown in Table 4.
Table 4. Project Activity Crashing Costs.
Planned Crash
Activity Description Duration Cost Duration Cost Critical Path? D cost
D
time (xi) r = D cost/D time
A Start 1 $ 1,580 0.5 $ 1,462 Yes $ 119 0.5 237.00
B Locate Facility 8 $ 13,456 6 $ 18,670 Yes $ 5,214 2 2607.10
C Order Furniture 9 $ 11,400 7 $ 16,403 Yes $ 5,003 2 2501.67
D Interview 10 $ 18,800 8 $ 27,824 No $ 9,024 2 4512.00
E Hire & Train 12 $ 28,880 8 $ 35,619 No $ 6,739 4 1684.67
F Remodel 9 $ 20,576 7 $ 29,607 No $ 9,031 2 4515.29
G Furniture Setup 8 $ 16,256 6 $ 22,555 Yes $ 6,299 2 3149.60
H Inspect 8 $ 24,064 7 $ 38,954 Yes $ 14,890 1 14889.60
I Occupy 8 $ 33,360 5 $ 38,573 Yes $ 5,213 3 1737.50
J Start Operations 1 $ 5,616 0.5 $ 5,195 Yes $ 421 0.5 842.40
Totals $ 173,988 $ 234,860
Students are shown how to formulate the problems using LP methods. The problem is to
minimize the cost of crashing subject to the constraints that the project duration must be less than
or equal to the new desired project completion date of 40 days.
Objective Function:
Min Crash Cost = 237.00Xa + 2607.10Xb + 2501.67Xc + 4512.00Xd + 1684.67Xe + 4515.29Xf +
3149.60Xg + 14889.60Xh + 1737.50Xi + 9547.20Xj
Subject to:
Maximum reduction constraints: Xa <= 1, Xb <= 2, Xc <= 2, Xd <= 2, Xe <= 4, Xf <= 2, Xg <= 2,
Xh <= 1, Xi <= 3, Xj <= 1
Non Negativity Constraints: Xi >= 0, Yi >=0, YFinish >=0
Start Constraints: Yb - Ya+ Xa >= 1 Yc - Yb + Xb >= 8 Yd -Yb + Xb >= 8
Ye - Yd + Xd >= 10 Yf - Yb + Xb >= 8 Yg -Yc + Xc >= 9 Yh - Ye + Xe >= 12
Yh - Yf + Xf >= 9 Yh - Yg + Xg >= 8 Yi - Yh + Xh >= 8 Yj - Yi + Xi >= 8
YFin - Yj + Xj >= 1 YFinish <= 40
Figure 5. LP Formulation to Minimize Cost of Crashing in Excel.
Figure 6. LP Formulation to Minimize Crashing Cost in Excel Solver.
Using LP to crash the project by three working days results in shortening activity A by 0.5 days,
activity E by 4 days, activity G by 0.5 days and activity I by 3 days with a project completion of
40 work days and a new cost of $192,120.
Table 5. Cost of Crashing the Project Duration to 40 Days.
Planned Crashed
Activity Description Duration Cost Duration Cost Critical Path? D cost
D
time (xi) r = D cost/D time
A Start 1 $ 1,580 0.5 $ 1,462 Yes $ 119 0.50 237.00
B Locate Facility 8 $ 13,456 6 $ 13,456 Yes $ - 0.00 0.00
C Order Furniture 9 $ 11,400 7 $ 11,400 Yes $ - 0.00 0.00
D Interview 10 $ 18,800 8 $ 18,800 No $ - 0.00 0.00
E Hire & Train 12 $ 28,880 8 $ 35,619 No $ 6,739 4.00 1684.67
F Remodel 9 $ 20,576 7 $ 20,576 No $ - 0.00 0.00
G Furniture Setup 8 $ 16,256 6 $ 22,555 Yes $ 6,299 0.50 12598.40
H Inspect 8 $ 24,064 7 $ 24,064 Yes $ - 0.00 0.00
I Occupy 8 $ 33,360 5 $ 38,573 Yes $ 5,213 3.00 1737.50
J Start Operations 1 $ 5,616 0.5 $ 5,616 Yes $ - 0.00 0.00
Total Cost $ 173,988 $ 192,120
The Activity List tab of the spreadsheet will automatically generate a Gantt chart showing the
planned schedule in blue and the crashed schedule in grey. The Gantt chart combined with the
cost tables provide a way to visually display the results of planning and can easily support what-
if analysis to determine schedule and cost for various scope changes.
Figure 7. Project Gantt Chart for Schedule Options.
Usage
The integrated spreadsheet model presented in this paper is a useful tool for professors teaching
schedule development, resource leveling and project crashing. Spreadsheets are readily available
to both students and faculty and our integrated example uses standard excel functions without
macros. The model is easily modified to expand additional tasks and incorporate more complex
examples by using a few copy commands. While students are informed of the math behind the
calculations the spreadsheet only requires the student to input required data and the model is
generally set up to auto calculate and produce the output and charts for analysis. This allows
students to concentrate on properly describing the problem and analyzing the solution while not
having to perform tedious calculations. We have used our model in both undergraduate and
graduate introductory project management courses to reinforce key concepts and broaden student
understanding of not just developing a schedule but managing it. The responses from students
have been favorable as most students like the ease of use and do not have to deal with the
learning another software program. They also state that the integrated exercises make it easier for
them to understand the relationships between schedule, resources and requirements. Prior to the
exercise faculty covered schedule development, resource allocation and project crashing as
separate lecture topics. The integrated exercise has allowed us to spend less time on each of the
individual topics and more time demonstrating through practical application how these topics
overlap and influence each other. For future research we are planning to perform before and after
analysis of grades for the exercise in order to measure any improvement in student performance
after completing the integrated exercise problem. We are planning to introduce a simple exercise
in which students will manually build a schedule, assign resources and crash the schedule
followed by the integrated spreadsheet exercise. The majority of undergraduate students are
engineering majors while less than 50% of our graduate students have engineering degrees. For
the graduate students we have been able to compare two classes, pre-introduction and post-
introduction. The average score for our post-introduction facility Relocation Exercise class were
8% higher than the pre-introduction class.
Conclusions
Integrated spreadsheet models are an effective means of enforcing key project management and
decision-making skills in students. Automating many of the analytical techniques allows students
to focus on understanding the concepts and applying that understanding to analyze potential
solutions to meet customer needs. The integrated exercise can also be used to show the students
how to pin-point the part of the project that if delayed beyond the allotted time would increase
the completion time of the project as a whole. It further assists in allocating resources and thus
helps to make the total cost of the project a minimum by finding the optimum balance between
various costs and time involved. This paper describes one approach to expanding student
comprehension of schedule development and management using an integrated spreadsheet
example based on built in functions and formulas that support sensitivity analysis for decision
making. As student feedback on using the example has been positive we intend to continue its
use and expand the scenario to include a more complex schedule.
References
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linear programming: case study. International journal of scientific & technology research, 4(8),
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[2] Arcuri, F. J., & Hildreth, J. C. (2007). The principles of schedule impact analysis. VDOT-VT
Partnership for Project Scheduling, Blacksburg, VA.
[3] Hildreth, J. C. and Munoz, B. P. (2005). An Introduction to the Management Principles of
Scheduling, TR-05-04, A report presented to the Virginia Department of Transportation and the
VDOT-VT Partnership for Project Scheduling Advisory Board.
[4] Seal, K. (2001) A generalized PERT/CPM implementation in a spreadsheet. INFORMS
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Spreadsheets in Education (eJSiE): Vol. 1: Iss.1, Article 2.
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