Introduction
Similar to the vignette on Bayesian networks for project risk analysis, we will use Bayesian Networks to analyze project portfolio risks. Bayesian Networks are probabilistic graphical models that represent a set of variables and their conditional dependencies via a directed acyclic graph (DAG). Bayesian Networks are useful for modeling complex systems and making inferences about the relationships between variables.
Project Portfolio Risk Analysis
Suppose we have a simple project portfolio. The portfolio consists of 3 civil engineering projects: construction of a roadway, a small building construction, and a pedestrian bridge project. The primary tasks for these projects are as follows.
roadway_tasks <- data.frame(
ID = c("AF", "AG", "AH", "AI", "AJ", "AK", "AL", "AM"),
Label = c(
"Task-1.1",
"Task-1.2",
"Task-1.3",
"Task-1.4",
"Task-1.5",
"Task-1.6",
"Task-1.7",
"Task-1.8"
),
Task = c(
"Survey and Site Assessment",
"Design and Planning",
"Permitting and Approvals",
"Excavation and Grading",
"Pavement Installation",
"Drainage and Utilities Installation",
"Signage and Markings",
"Final Inspection and Handover"
),
Project_ID = rep("BF", 8)
)
knitr::kable(roadway_tasks, caption = "Project 1: Roadway Tasks")| ID | Label | Task | Project_ID |
|---|---|---|---|
| AF | Task-1.1 | Survey and Site Assessment | BF |
| AG | Task-1.2 | Design and Planning | BF |
| AH | Task-1.3 | Permitting and Approvals | BF |
| AI | Task-1.4 | Excavation and Grading | BF |
| AJ | Task-1.5 | Pavement Installation | BF |
| AK | Task-1.6 | Drainage and Utilities Installation | BF |
| AL | Task-1.7 | Signage and Markings | BF |
| AM | Task-1.8 | Final Inspection and Handover | BF |
building_tasks <- data.frame(
ID = c("AN", "AO", "AP", "AQ", "AR", "AS", "AT", "AU"),
Label = c(
"Task-2.1",
"Task-2.2",
"Task-2.3",
"Task-2.4",
"Task-2.5",
"Task-2.6",
"Task-2.7",
"Task-2.8"
),
Task = c(
"Architectural Design",
"Structural Engineering",
"Regulatory Approvals",
"Foundation and Excavation",
"Framing and Structural Work",
"Plumbing, Electrical, and HVAC Install",
"Interior and Exterior Finishing",
"Final Inspection and Handover"
),
Project_ID = rep("BG", 8)
)
knitr::kable(building_tasks, caption = "Project 2: Building Tasks")| ID | Label | Task | Project_ID |
|---|---|---|---|
| AN | Task-2.1 | Architectural Design | BG |
| AO | Task-2.2 | Structural Engineering | BG |
| AP | Task-2.3 | Regulatory Approvals | BG |
| AQ | Task-2.4 | Foundation and Excavation | BG |
| AR | Task-2.5 | Framing and Structural Work | BG |
| AS | Task-2.6 | Plumbing, Electrical, and HVAC Install | BG |
| AT | Task-2.7 | Interior and Exterior Finishing | BG |
| AU | Task-2.8 | Final Inspection and Handover | BG |
bridge_tasks <- data.frame(
ID = c("AV", "AW", "AX", "AY", "AZ", "BA", "BB", "BC", "BD", "BE"),
Label = c(
"Task-3.1",
"Task-3.2",
"Task-3.3",
"Task-3.4",
"Task-3.5",
"Task-3.6",
"Task-3.7",
"Task-3.8",
"Task-3.9",
"Task-3.10"
),
Task = c(
"Site Survey and Assessment",
"Environmental Impact Study",
"Concept Design and Planning",
"Structural Engineering and Analysis",
"Permitting and Approvals",
"Foundation and Pile Installation",
"Superstructure Construction",
"Decking and Surface Finishing",
"Inspection and Load Testing",
"Final Handover"
),
Project_ID = rep("BH", 10)
)
knitr::kable(bridge_tasks, caption = "Project 3: Bridge Tasks")| ID | Label | Task | Project_ID |
|---|---|---|---|
| AV | Task-3.1 | Site Survey and Assessment | BH |
| AW | Task-3.2 | Environmental Impact Study | BH |
| AX | Task-3.3 | Concept Design and Planning | BH |
| AY | Task-3.4 | Structural Engineering and Analysis | BH |
| AZ | Task-3.5 | Permitting and Approvals | BH |
| BA | Task-3.6 | Foundation and Pile Installation | BH |
| BB | Task-3.7 | Superstructure Construction | BH |
| BC | Task-3.8 | Decking and Surface Finishing | BH |
| BD | Task-3.9 | Inspection and Load Testing | BH |
| BE | Task-3.10 | Final Handover | BH |
Resources
The primary resources required for these projects are as follows.
roadway_resources <- data.frame(
ID = c("F", "G", "H", "I", "J", "K", "L", "M"),
Label = c(
"Resource-1.1",
"Resource-1.2",
"Resource-1.3",
"Resource-1.4",
"Resource-1.5",
"Resource-1.6",
"Resource-1.7",
"Resource-1.8"
),
Resource = c(
"Surveyer",
"Engineer",
"Regulatory Support",
"Heavy Machinery",
"Pavement and Related Machinery",
"Drainage Material and Equipment",
"Painters, Traffic Signs, Road Markers",
"Inspectors and Quality Control Support"
),
Task_ID = c("AF", "AG", "AH", "AI", "AJ", "AK", "AL", "AM"),
Task = c(
"Survey and Site Assessment",
"Design and Planning",
"Permitting and Approvals",
"Excavation and Grading",
"Pavement Installation",
"Drainage and Utilities Installation",
"Signage and Markings",
"Final Inspection and Handover"
),
Mean = c(
10000,
20000,
3500,
35000,
100000,
25000,
6500,
2000
),
SD = c(
2000,
5000,
1000,
10000,
20000,
5000,
1500,
500
)
)
knitr::kable(roadway_resources, caption = "Project 1: Roadway Resources")| ID | Label | Resource | Task_ID | Task | Mean | SD |
|---|---|---|---|---|---|---|
| F | Resource-1.1 | Surveyer | AF | Survey and Site Assessment | 10000 | 2000 |
| G | Resource-1.2 | Engineer | AG | Design and Planning | 20000 | 5000 |
| H | Resource-1.3 | Regulatory Support | AH | Permitting and Approvals | 3500 | 1000 |
| I | Resource-1.4 | Heavy Machinery | AI | Excavation and Grading | 35000 | 10000 |
| J | Resource-1.5 | Pavement and Related Machinery | AJ | Pavement Installation | 100000 | 20000 |
| K | Resource-1.6 | Drainage Material and Equipment | AK | Drainage and Utilities Installation | 25000 | 5000 |
| L | Resource-1.7 | Painters, Traffic Signs, Road Markers | AL | Signage and Markings | 6500 | 1500 |
| M | Resource-1.8 | Inspectors and Quality Control Support | AM | Final Inspection and Handover | 2000 | 500 |
building_resources <- data.frame(
ID = c("N", "O", "P", "Q", "R", "S", "T", "U"),
Label = c(
"Resource-2.1",
"Resource-2.2",
"Resource-2.3",
"Resource-2.4",
"Resource-2.5",
"Resource-2.6",
"Resource-2.7",
"Resource-2.8"
),
Resource = c(
"Architect",
"Structural Engineer",
"Regulatory Support",
"Heavy Machinery",
"Building Materials",
"Plumbers, Electricians",
"Painters, Interior Finishers",
"Inspector and Quality Control Support"
),
Task_ID = c("AN", "AO", "AP", "AQ", "AR", "AS", "AT", "AU"),
Task = c(
"Architectural Design",
"Structural Engineering",
"Regulatory Approvals",
"Foundation and Excavation",
"Framing and Structural Work",
"Plumbing, Electrical, and HVAC Install",
"Interior and Exterior Finishing",
"Final Inspection and Handover"
),
Mean = c(
15000,
30000,
4000,
40000,
100000,
20000,
8000,
2500
),
SD = c(
3000,
6000,
1000,
10000,
20000,
4000,
1500,
500
)
)
knitr::kable(building_resources, caption = "Project 2: Building Resources")| ID | Label | Resource | Task_ID | Task | Mean | SD |
|---|---|---|---|---|---|---|
| N | Resource-2.1 | Architect | AN | Architectural Design | 15000 | 3000 |
| O | Resource-2.2 | Structural Engineer | AO | Structural Engineering | 30000 | 6000 |
| P | Resource-2.3 | Regulatory Support | AP | Regulatory Approvals | 4000 | 1000 |
| Q | Resource-2.4 | Heavy Machinery | AQ | Foundation and Excavation | 40000 | 10000 |
| R | Resource-2.5 | Building Materials | AR | Framing and Structural Work | 100000 | 20000 |
| S | Resource-2.6 | Plumbers, Electricians | AS | Plumbing, Electrical, and HVAC Install | 20000 | 4000 |
| T | Resource-2.7 | Painters, Interior Finishers | AT | Interior and Exterior Finishing | 8000 | 1500 |
| U | Resource-2.8 | Inspector and Quality Control Support | AU | Final Inspection and Handover | 2500 | 500 |
bridge_resources <- data.frame(
ID = c("V", "W", "X", "Y", "Z", "AA", "AB", "AC", "AD", "AE"),
Label = c(
"Resource-3.1",
"Resource-3.2",
"Resource-3.3",
"Resource-3.4",
"Resource-3.5",
"Resource-3.6",
"Resource-3.7",
"Resource-3.8",
"Resource-3.9",
"Resource-3-10"
),
Resource = c(
"Surveyor",
"Environmental Scientist",
"Civil Engineer",
"Structural Engineer",
"Regulatory Consulting",
"Heavy Machinery",
"Steel and Concrete Materials",
"Decking Materials",
"Inspector and Quality Control Support",
"Handover Team"
),
Task_ID = c("AV", "AW", "AX", "AY", "AZ", "BA", "BB", "BC", "BD", "BE"),
Task = c(
"Site Survey and Assessment",
"Environmental Impact Study",
"Concept Design and Planning",
"Structural Engineering and Analysis",
"Permitting and Approvals",
"Foundation and Pile Installation",
"Superstructure Construction",
"Decking and Surface Finishing",
"Inspection and Load Testing",
"Final Handover"
),
Mean = c(
10000,
20000,
30000,
40000,
5000,
50000,
100000,
20000,
5000,
10000
),
SD = c(
2000,
4000,
6000,
8000,
1000,
10000,
20000,
4000,
1000,
2000
)
)
knitr::kable(bridge_resources, caption = "Project 3: Bridge Resources")| ID | Label | Resource | Task_ID | Task | Mean | SD |
|---|---|---|---|---|---|---|
| V | Resource-3.1 | Surveyor | AV | Site Survey and Assessment | 1e+04 | 2000 |
| W | Resource-3.2 | Environmental Scientist | AW | Environmental Impact Study | 2e+04 | 4000 |
| X | Resource-3.3 | Civil Engineer | AX | Concept Design and Planning | 3e+04 | 6000 |
| Y | Resource-3.4 | Structural Engineer | AY | Structural Engineering and Analysis | 4e+04 | 8000 |
| Z | Resource-3.5 | Regulatory Consulting | AZ | Permitting and Approvals | 5e+03 | 1000 |
| AA | Resource-3.6 | Heavy Machinery | BA | Foundation and Pile Installation | 5e+04 | 10000 |
| AB | Resource-3.7 | Steel and Concrete Materials | BB | Superstructure Construction | 1e+05 | 20000 |
| AC | Resource-3.8 | Decking Materials | BC | Decking and Surface Finishing | 2e+04 | 4000 |
| AD | Resource-3.9 | Inspector and Quality Control Support | BD | Inspection and Load Testing | 5e+03 | 1000 |
| AE | Resource-3-10 | Handover Team | BE | Final Handover | 1e+04 | 2000 |
Risks
The primary risks for these projects are as follows.
roadway_risks <- data.frame(
Risk_ID = c("A", "B", "C"),
Name = c(
"Risk-1",
"Risk-2",
"Risk-3"
),
Risk = c(
"Delays in Permitting and Approvals",
"Unforeseen Site Conditions",
"Material Price Fluctuations"
),
Probability = c(
0.9,
0.95,
0.8
),
Resource_ID = c("H", "I", "J"),
Resource_Impacted = c(
"Regulatory Support",
"Heavy Machinery",
"Pavement and Related Machinery"
),
Mean = c(
7000,
70000,
200000
),
SD = c(
2000,
20000,
40000
)
)
knitr::kable(roadway_risks, caption = "Project 1: Roadway Risks")| Risk_ID | Name | Risk | Probability | Resource_ID | Resource_Impacted | Mean | SD |
|---|---|---|---|---|---|---|---|
| A | Risk-1 | Delays in Permitting and Approvals | 0.90 | H | Regulatory Support | 7e+03 | 2000 |
| B | Risk-2 | Unforeseen Site Conditions | 0.95 | I | Heavy Machinery | 7e+04 | 20000 |
| C | Risk-3 | Material Price Fluctuations | 0.80 | J | Pavement and Related Machinery | 2e+05 | 40000 |
building_risks <- data.frame(
Risk_ID = c("A", "D", "C"),
Name = c(
"Risk-1",
"Risk-4",
"Risk-3"
),
Risk = c(
"Delays in Permitting and Approvals",
"Labor Shortage or Skills Gap",
"Material Price Volatility"
),
Probability = c(
0.9,
0.9,
0.8
),
Resource_ID = c("P", "S", "O"),
Resource = c(
"Regulatory Support",
"Plumbers, Electricians",
"Building Materials"
),
Mean = c(
8000,
40000,
60000
),
SD = c(
2000,
8000,
12000
)
)
knitr::kable(building_risks, caption = "Project 2: Building Risks")| Risk_ID | Name | Risk | Probability | Resource_ID | Resource | Mean | SD |
|---|---|---|---|---|---|---|---|
| A | Risk-1 | Delays in Permitting and Approvals | 0.9 | P | Regulatory Support | 8000 | 2000 |
| D | Risk-4 | Labor Shortage or Skills Gap | 0.9 | S | Plumbers, Electricians | 40000 | 8000 |
| C | Risk-3 | Material Price Volatility | 0.8 | O | Building Materials | 60000 | 12000 |
bridge_risks <- data.frame(
Risk_ID = c("B", "D", "E"),
Name = c(
"Risk-2",
"Risk-4",
"Risk-5"
),
Risk = c(
"Unforeseen Environmental Conditions",
"Labor Supply Disruptions",
"Structural Design Revisions"
),
Probabiliy = c(
0.95,
0.9,
0.95
),
Resource_ID = c("W", "AD", "Y"),
Resource = c(
"Environmental Scientist",
"Inspector and Quality Control Support",
"Structural Engineer"
),
Mean = c(
40000,
10000,
80000
),
SD = c(
9000,
2000,
16000
)
)
knitr::kable(bridge_risks, caption = "Project 3: Bridge Risks")| Risk_ID | Name | Risk | Probabiliy | Resource_ID | Resource | Mean | SD |
|---|---|---|---|---|---|---|---|
| B | Risk-2 | Unforeseen Environmental Conditions | 0.95 | W | Environmental Scientist | 40000 | 9000 |
| D | Risk-4 | Labor Supply Disruptions | 0.90 | AD | Inspector and Quality Control Support | 10000 | 2000 |
| E | Risk-5 | Structural Design Revisions | 0.95 | Y | Structural Engineer | 80000 | 16000 |
Notice how most of the risks are common across multiple projects. Key portfolo-level risks are Regulatory Delays, Unforeseen Site Conditions, Material Price Volatility, and Labor/Supply Disruptions.
Bayesian Network
To model the project portfolio risks, we will use a Bayesian Network. The Bayesian Network will represent the dependencies between the projects, tasks, resources, and risks. The Bayesian Network will help us understand the relationships between the variables and make inferences about the risks.
First, we need to define the nodes and edges of the Bayesian network. The nodes represent the variables, and the edges represent the dependencies between the variables. The following code defines the nodes and edges of the Bayesian network.
nodes <- data.frame(
id = c("A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N",
"O", "P", "Q", "R", "S", "T", "U", "V", "W", "X", "Y", "Z", "AA", "AB",
"AC", "AD", "AE", "AF", "AG", "AH", "AI", "AJ", "AK", "AL", "AM", "AN",
"AO", "AP", "AQ", "AR", "AS", "AT", "AU", "AV", "AW", "AX", "AY", "AZ",
"BA", "BB", "BC", "BD", "BE", "BF", "BG", "BH", "BI"),
label = c(
"Risk-1",
"Risk-2",
"Risk-3",
"Risk-4",
"Risk-5",
"Resource-1.1",
"Resource-1.2",
"Resource-1.3",
"Resource-1.4",
"Resource-1.5",
"Resource-1.6",
"Resource-1.7",
"Resource-1.8",
"Resource-2.1",
"Resource-2.2",
"Resource-2.3",
"Resource-2.4",
"Resource-2.5",
"Resource-2.6",
"Resource-2.7",
"Resource-2.8",
"Resource-3.1",
"Resource-3.2",
"Resource-3.3",
"Resource-3.4",
"Resource-3.5",
"Resource-3.6",
"Resource-3.7",
"Resource-3.8",
"Resource-3.9",
"Resource-3-10",
"Task-1.1",
"Task-1.2",
"Task-1.3",
"Task-1.4",
"Task-1.5",
"Task-1.6",
"Task-1.7",
"Task-1.8",
"Task-2.1",
"Task-2.2",
"Task-2.3",
"Task-2.4",
"Task-2.5",
"Task-2.6",
"Task-2.7",
"Task-2.8",
"Task-3.1",
"Task-3.2",
"Task-3.3",
"Task-3.4",
"Task-3.5",
"Task-3.6",
"Task-3.7",
"Task-3.8",
"Task-3.9",
"Task-3.10",
"Project 1",
"Project 2",
"Project 3",
"Project Portfolio"
),
stringsAsFactors = FALSE
)Next, we define the edges of the Bayesian network. The following code defines the edges of the Bayesian network.
links <- data.frame(
source = c("A", "B", "C", "A", "D", "C", "B", "D", "E", "F", "G", "H", "I",
"J", "K", "L", "M", "N", "O", "P", "Q", "R", "S", "T", "U", "V", "W",
"X", "Y", "Z", "AA", "AB", "AC", "AD", "AE", "AF", "AG", "AH", "AI",
"AJ", "AK", "AL", "AM", "AN", "AO", "AP", "AQ", "AR", "AS", "AT", "AU",
"AV", "AW", "AX", "AY", "AZ", "BA", "BB", "BC", "BD", "BE", "BF", "BG",
"BH"),
target = c("H", "I", "J", "P", "S", "O", "W", "AD", "Y", "AF", "AG", "AH", "AI",
"AJ", "AK", "AL", "AM", "AN", "AO", "AP", "AQ", "AR", "AS", "AT", "AU",
"AV", "AW", "AX", "AY", "AZ", "BA", "BB", "BC", "BD", "BE", "BF", "BF",
"BF", "BF", "BF", "BF", "BF", "BF", "BG", "BG", "BG", "BG", "BG", "BG",
"BG", "BG", "BH", "BH", "BH", "BH", "BH", "BH", "BH", "BH", "BH", "BH",
"BI", "BI", "BI")
)Then we define the distributions for the nodes. The following code defines the distributions for the nodes.
distributions <- list(
A = list(
type = "discrete",
values = c(1, 0),
probs = c(0.9, 0.1)
),
B = list(
type = "discrete",
values = c(1, 0),
probs = c(0.95, 0.05)
),
C = list(
type = "discrete",
values = c(1, 0),
probs = c(0.8, 0.2)
),
D = list(
type = "discrete",
values = c(1, 0),
probs = c(0.9, 0.1)
),
E = list(
type = "discrete",
values = c(1, 0),
probs = c(0.95, 0.05)
),
F = list(
type = "normal",
mean = 10000,
sd = 2000
),
G = list(
type = "normal",
mean = 20000,
sd = 5000
),
H = list(
type = "conditional",
condition = c("A"),
true_dist = list(
type = "normal",
mean = 7000,
sd = 2000
),
false_dist = list(
type = "normal",
mean = 3500,
sd = 1000
)
),
I = list(
type = "conditional",
condition = c("B"),
true_dist = list(
type = "normal",
mean = 70000,
sd = 20000
),
false_dist = list(
type = "normal",
mean = 35000,
sd = 10000
)
),
J = list(
type = "conditional",
condition = c("C"),
true_dist = list(
type = "normal",
mean = 100000,
sd = 40000
),
false_dist = list(
type = "normal",
mean = 50000,
sd = 20000
)
),
K = list(
type = "normal",
mean = 25000,
sd = 5000
),
L = list(
type = "normal",
mean = 6500,
sd = 1500
),
M = list(
type = "normal",
mean = 2000,
sd = 500
),
N = list(
type = "normal",
mean = 15000,
sd = 3000
),
O = list(
type = "conditional",
condition = c("C"),
true_dist = list(
type = "normal",
mean = 8000,
sd = 2000
),
false_dist = list(
type = "normal",
mean = 4000,
sd = 1000
)
),
P = list(
type = "conditional",
condition = c("A"),
true_dist = list(
type = "normal",
mean = 8000,
sd = 2000
),
false_dist = list(
type = "normal",
mean = 4000,
sd = 1000
)
),
Q = list(
type = "normal",
mean = 40000,
sd = 10000
),
R = list(
type = "normal",
mean = 100000,
sd = 20000
),
S = list(
type = "conditional",
condition = c("D"),
true_dist = list(
type = "normal",
mean = 16000,
sd = 4000
),
false_dist = list(
type = "normal",
mean = 8000,
sd = 2000
)
),
T = list(
type = "normal",
mean = 8000,
sd = 1500
),
U = list(
type = "normal",
mean = 2500,
sd = 500
),
V = list(
type = "normal",
mean = 10000,
sd = 2000
),
W = list(
type = "conditional",
condition = c("B"),
true_dist = list(
type = "normal",
mean = 60000,
sd = 4000
),
false_dist = list(
type = "normal",
mean = 30000,
sd = 2000
)
),
X = list(
type = "normal",
mean = 30000,
sd = 6000
),
Y = list(
type = "conditional",
condition = c("E"),
true_dist = list(
type = "normal",
mean = 20000,
sd = 2000
),
false_dist = list(
type = "normal",
mean = 10000,
sd = 1000
)
),
Z = list(
type = "normal",
mean = 5000,
sd = 1000
),
AA = list(
type = "normal",
mean = 50000,
sd = 10000
),
AB = list(
type = "normal",
mean = 100000,
sd = 20000
),
AC = list(
type = "normal",
mean = 20000,
sd = 4000
),
AD = list(
type = "conditional",
condition = c("D"),
true_dist = list(
type = "normal",
mean = 40000,
sd = 8000
),
false_dist = list(
type = "normal",
mean = 20000,
sd = 4000
)
),
AE = list(
type = "normal",
mean = 10000,
sd = 2000
),
AF = list(
type = "aggregate",
nodes = c("F")
),
AG = list(
type = "aggregate",
nodes = c("G")
),
AH = list(
type = "aggregate",
nodes = c("H")
),
AI = list(
type = "aggregate",
nodes = c("I")
),
AJ = list(
type = "aggregate",
nodes = c("J")
),
AK = list(
type = "aggregate",
nodes = c("K")
),
AL = list(
type = "aggregate",
nodes = c("L")
),
AM = list(
type = "aggregate",
nodes = c("M")
),
AN = list(
type = "aggregate",
nodes = c("N")
),
AO = list(
type = "aggregate",
nodes = c("O")
),
AP = list(
type = "aggregate",
nodes = c("P")
),
AQ = list(
type = "aggregate",
nodes = c("Q")
),
AR = list(
type = "aggregate",
nodes = c("R")
),
AS = list(
type = "aggregate",
nodes = c("S")
),
AT = list(
type = "aggregate",
nodes = c("T")
),
AU = list(
type = "aggregate",
nodes = c("U")
),
AV = list(
type = "aggregate",
nodes = c("V")
),
AW = list(
type = "aggregate",
nodes = c("W")
),
AX = list(
type = "aggregate",
nodes = c("X")
),
AY = list(
type = "aggregate",
nodes = c("Y")
),
AZ = list(
type = "aggregate",
nodes = c("Z")
),
BA = list(
type = "aggregate",
nodes = c("AA")
),
BB = list(
type = "aggregate",
nodes = c("AB")
),
BC = list(
type = "aggregate",
nodes = c("AC")
),
BD = list(
type = "aggregate",
nodes = c("AD")
),
BE = list(
type = "aggregate",
nodes = c("AE")
),
BF = list(
type = "aggregate",
nodes = c("AF", "AG", "AH", "AI", "AJ", "AK", "AL", "AM")
),
BG = list(
type = "aggregate",
nodes = c("AN", "AO", "AP", "AQ", "AR", "AS", "AT", "AU")
),
BH = list(
type = "aggregate",
nodes = c("AV", "AW", "AX", "AY", "AZ", "BA", "BB", "BC", "BD", "BE")
),
BI = list(
type = "aggregate",
nodes = c("BF", "BG", "BH")
)
)Finally, we define the Bayesian network using the nodes, edges, and distributions. The following code defines the Bayesian network.
To plot the Bayesian network, we can use the igraph package. The igraph package provides functions for creating and analyzing graph structures. The following code plots the Bayesian network.
library(igraph)
g <- graph_from_data_frame(graph$links, vertices = graph$nodes, directed = TRUE)
plot(g, main = "Bayesian Network", vertex.label = graph$nodes$label,
vertex.size = 5, vertex.color = "lightblue", edge.arrow.size = 0.5,
edge.color = "gray", layout = layout_with_fr)
The Bayesian network represents the dependencies between the projects, tasks, resources, and risks. The Bayesian network helps us understand the relationships between the variables and make inferences about the risks.
Inference
To make inferences about the risks, we can use probabilistic inference. Probabilistic inference is the process of estimating the probability distribution of a variable given evidence about other variables. Probabilistic inference allows us to make predictions about the risks based on the observed data.
simulation_results <- prob_net_sim(graph, num_samples = 1000)We can use these results to estimate the total project cost and assess the impact of risks on the project outcomes.
hist <- hist(simulation_results$BI, breaks = 50, plot = FALSE)
plot(hist, main = "Total Project Portfolio Cost", xlab = "Project Cost", col = "skyblue", border = "white")
The histogram shows the distribution of the total project cost. The histogram helps us understand the range of possible project costs and assess the impact of risks on the project outcomes.
Learning
The Bayesian network can be updated with new data to improve the model’s accuracy. Learning and updating the Bayesian network involves updating the distributions based on new evidence. The updated Bayesian network can be used to make more accurate predictions about the risks.
For example, suppose we have new data that Risk 2 occurred. We can update the Bayesian network with this new data. The following code updates the Bayesian network with the new data.
updated_results <- prob_net_learn(graph, observations = list(B = "Yes"),
num_samples = 1000)We can use the updated results to estimate the total project cost and assess the impact of risks on the project outcomes.
hist <- hist(simulation_results$I, breaks = 50, plot = FALSE)
hist2 <- hist(updated_results$I, breaks = 50, plot = FALSE)
plot(hist, main = "Heavy Machinery Resource Cost", xlab = "Resource Cost",
col = "skyblue", border = "white")
plot(hist2, col = "blue", border = "white", add = TRUE)
legend("topright", legend = c("Before Update", "After Update"),
fill = c("skyblue", "blue"), bty = "n")
Updating
The Bayesian network can also be updated by adding or removing arcs between nodes. Adding or removing arcs can change the dependencies between the variables and improve the model’s accuracy.
For example, suppose we learn that Resource 1.3 is no longer at risk of delays. We can remove the arc between Resource 1.3 and Risk 1.1. The following code removes the arc between Resource 1.3 and Risk 1.1.
remove_links <- data.frame(
source = c("A"),
target = c("H"),
stringsAsFactors = FALSE
)
update_distributions <- list(
H = list(
type = "normal",
mean = 3500,
sd = 1000
)
)
updated_graph <- prob_net_update(graph, remove_links = remove_links,
update_distributions = update_distributions)
updated_results <- prob_net_sim(updated_graph, num_samples = 1000)We can compare the results before and after updating the Bayesian network. The following code compares the results before and after updating the Bayesian network.
hist <- hist(simulation_results$H, breaks = 50, plot = FALSE)
hist2 <- hist(updated_results$H, breaks = 50, plot = FALSE)
plot(hist, main = "Regulatory Support Resource Cost", xlab = "Resource Cost",
col = "skyblue", border = "white", ylim = c(0, max(hist$counts, hist2$counts)))
plot(hist2, col = "blue", border = "white", add = TRUE)
legend("topright", legend = c("Before Update", "After Update"),
fill = c("skyblue", "blue"), bty = "n")
Conclusion
In this vignette, we used Bayesian Networks to analyze project portfolio risks. We defined the projects, tasks, resources, and risks for a simple project portfolio. We then defined a Bayesian network to represent the dependencies between the variables. We used probabilistic inference to make inferences about the risks and estimate the total project cost. We also learned and updated the Bayesian network with new data to improve the model’s accuracy. The Bayesian network provides a powerful tool for modeling project portfolio risks and making informed decisions about project outcomes.