Introduction

This is an R program that takes an activity table whether it is a CPM or a PERT and and proccess it to find the critical path (along with some other calculations in the PERT method )

The Project Evaluation and Review Technique (PERT) and Critical Path Method (CPM) were developed by management scientists to help organizations with planning, scheduling and controlling large projects,such as building a new hospital orlaunching a new product.

large projects has series of interdependent activities that take time to complete,require funds and resources, such as time andlabor.

Interdependence means that activities follow a given sequence orprecedence relationship -some activities cannot start until others are completed.

Steps :

• Step 1. Identify theactivities
  
• Step 2. Determine activity relationships (immediate predecessors of eachactivity)
  
• Step 3. Estimate activity completion times andcosts
  
• Step 4. Construct an activitynetwork
  
• Step 5. Execute a forward pass to determine earliest start and earliest finish times for each activity, and project completion time
  
• Step 6. Execute a backward pass to determine latest start and latest finish times for each activity
  
• Step 7. Identify activity slack (length of time an activity can be delayed without delaying the project completiontime)
  
• Step 8. Find the activities with zero slack; these are critical activities and make up at least one critical path
  
• Step 9. Use information from Steps 5 -8 to develop the activity schedule for the project.
  
• Step 10. Find project completion time variance and conduct probability analysis, suchas the probability of meeting a customer target completion time, under the condition of uncertainty in activity times.
  
• Step 11. Consider time-cost tradeoffs
  
• Step 10. Implement, monitor and control the project.
  

1- Critical Path Method

The critical path method, or critical path analysis, is an algorithm for scheduling a set of project activities. It is commonly used in conjunction with the program evaluation and review technique.

Steps :

• takes an activity table with each activity's titel, immediate predecessor(s) and time and stores it into a graph

• using a modified Breadth First Search algorithm to traverse the graph to find the forward and backward path

• in the forward path it calculates the ES (Early Start time) and EF (Early Finish time ) of each activity

• in the backward path it calculates the LS (Late Start time ) and LF (Late Finish time ) of each activity

• calculate the slack value of each activity

  ES (Early Start Time) = maximum value of "EF" for all the activities it depends on
  EF (Early Finish Time) = ES + Activity time
  LF (Late Finish Time) = minimum value of "LS" for all the activites that depends on this activity LS (Late Start Time ) = LF - Activity time
  S (Slack Value) = LS - ES of each activity, if it's zero then this activity is in the critical path

• draws a plot showing the graph, the attributes for each node and the critical path (green nodes),

Here is Some Sample Tests And It's Output

Activity	Title	Immediate Predecessors	Activity Time
A	Identify market need	-	3
B	Conduct R&D	A	60
C	Design packaging	A	5
D	Select product size	A	15
E	Conduct product test	B	6
F	Install production process	D	40
G	Perform market analysis	C E	10
H	Production startup	F	7
I	Make modifications	G	6
J	Market product	H I	12

   

---

   

Activity	Title	Immediate Predecessor	Activity Time
A	High-level analysis	-	7
B	Selection of hardware platform	A	7
C	Installation and commissioning of hardware	B	14
D	Detailed analysis of core modules	A	14
E	Detailed analysis of supporting modules	D	14
F	Programming of core modules	D	14
G	Programming of supporting modules	E	21
H	Quality assurance of core modules	F	7
I	Quality assurance of supporting modules	G	7
J	Core module training	C H	1
K	Development and QA of accounting reporting	E	7
L	Development and QA of management reporting	E	7
M	Development of Management Information System	L	7
N	Detailed training	I J K M	7

   

---

   

Activity	Title	Immeediate Predecessors	Activity Time
A	"TEST"	-	2
B	"TSET"	A	4
C	"TEST"	B	3
D	"TEST"	B	2
E	"TEST"	C D	10
F	"TEST"	B	4

   

---

   

Activity	Title	Immediate Predecessors	Activity Time
A	"TEST"	-	3
B	"TEST"	A	60
C	"TEST"	A	5
D	"TEST"	A	15
E	"TEST"	B	6
F	"TEST"	D	40
G	"TEST"	C E	10
H	"TEST"	F	7

---

---

2- PERT method

PERT is a scheduling technique that was specifically designed by the Navy in 1958for projects with uncertain activitytimes.

Steps :

• Takes an activities table as follow (Activity - Title - Immediate Predecessor - Optimistic Time - Most probable Time - Pessimistic Time )
• Finds the Expected Time for each activity by : expected_time(a) = ( Optimistic Time + 4 * Most probable Time + Pessimistic Time ) / 6
• Finds the variance of each activity by : variance(a) = (Pessimistic Time - Optimistic Time ) ^ 2 / 36
• sends the modified table to CPM function to find the critical path
• Find mean which is the sum of Expected Times of critical path activities
• Finds the total variance which is the sum of Variance o critical path activities
• Finds the standard deviation which is the square root of variance
• Finds the probabilty to finish the project on a specific Time

Here is Some Sample Tests And It's Output

Activity	Immediate Prodecessor	Optimistic Time	Most probable Time	Pessimistic Time
A	-	6	9	12
B	A	3	4	11
C	A	2	5	14
D	B-C	4	6	8
E	C	1	1.5	5
F	E	5	6	7
G	D	7	8	15
H	B	1	2	3

|              Task     |    ans     |
| -------------------   | ---------- |
|              Variance | 7.22222222 |
|    Standard Deviation | 2.68741925 |
| Propability on 26 day | 4.9%       |