1. WOOD 492 MODELLING FOR DECISION SUPPORT
Today we will talk about the format of the course and some introductions to operations research and its applications
Lecture 23
CPM

2. Review Minimum spanning tree
Goal: find the smallest network that has a path between each two nodes (application example: railway networks)
Greedy algorithm solution : start from any node and in each step pick the closest unconnected node and add it to the network
Critical path method (CPM) for time-cost trade-off
Goal: find the optimal plan to expedite some activities within a project in order to minimize the costs while meeting the project deadline
First, we visualize the activities within a project using a network (each activity is represented with a node in the network)
Then we find the “critical path”, which is the longest path through the project network, going from start to finish nodes.
Oct 31, 2012
Wood Saba Vahid

3. Example 15 – Project network
“Reliable Constructions Co.” has identified the activities within a plant construction project
The deadline is in 40 weeks
Project budget is $5.4 million
Based on activity information we create the project network
How long will the project take?
We need to find the paths through the project network and their lengths
Path: one of the routes following the arcs from Start to Finish node
Length: sum of the estimated durations of the activities along that path
Example 15
Oct 31, 2012
Wood Saba Vahid

4. Example 15 - Critical Path Method
There are six paths for this network
Start-A-B-C-D-G-H-M-Finish 40 weeks
Start-A-B-C-E-H-M-Finish 31 weeks
Start-A-B-C-E-F-J-K-N-Finish 43 weeks
Start-A-B-C-E-F-J-L-N-Finish 44 weeks
Start-A-B-C-I-J-K-N-Finish 41 weeks
Start-A-B-C-I-J-L-N-Finish 42 weeks
Activities on each path are done sequentially (can not overlap), so the project length can not be shorter than the length of a given path, but it might be longer
e.g. activity H has 2 predecessors (E, and G which is not on the path). Looking at the duration of activities after C, we see that activity E only takes 4 weeks, while activity D and G take 13 weeks combined. Since H needs to wait for both E and G to finish, the project will take much longer than the 31 weeks estimated length for the second path
Critical Path
Oct 31, 2012
Wood Saba Vahid

5. Critical Path Critical path: Start-A-B-C-E-F-J-L-N-Finish 44 weeks
The longest path through the project network
May have more than one critical path (same length)
Activities on the longest path can be done sequentially without having to wait for activities on other paths, so the project length is equal to the length of this path
All other paths will reach the Finish node sooner than the critical path
Project length is 44 wks: longer than the 40 wks deadline
The management should focus on reducing the length of activities on the critical path to finish the project by the deadline
Which activities to choose?
Oct 31, 2012
Wood Saba Vahid

6. CPM for time-cost trade-offs
Crashing an activity: using costly measured to reduce its length
CPM: determines how much (if any) to crash each activity to reduce the estimated duration of the project
Time-cost graph : shows the relationship between time and cost of an activity in the normal and crash modes
Assumptions:
Maximum time reduction and associated crash costs
For each activity can be estimated with certainty
Partially crashing an activity is possible, the associated
cost and time move along the line segment in the graph
Time-cost trade-off data are required for each activity
Cost
Crash cost
Crash
Normal
Normal cost
Crash time
Normal time
Duration
Oct 31, 2012
Wood Saba Vahid

7. Cost-time trade-off Example of cost-time trade-off for an activity
Activity J (putting up the wallboard)
It’s possible to reduce the duration of this activity by two weeks (through hiring temporary workers and using overtime)
Normal time: 8 weeks Normal Cost: $430,000
Crash time: 6 weeks Crash Cost:$490,000
Maximum reduction in time=8 – 6 = 2 weeks
Crash cost per week saved = (490,000 – 430,000)/2 = $30,000
So for each week the company saves in time, $30,000 are required in extra costs
Example 15
Oct 31, 2012
Wood Saba Vahid

8. Which activities to crash?
How to select the activities to crash, while minimizing the costs of doing so?
Marginal Cost analysis
Linear programming
Marginal Cost Analysis
select the longest path through the network
Among the activities on that path, select the one with the lowest “crash cost per week”
Reduce its length by one week if possible, if not move to the next lowest crash cost/week
Review the length of all paths after this reduction and repeat steps 1 to 3 until you reach the desired length of the project
Example 15
Oct 31, 2012
Wood Saba Vahid

9. Using LP to choose activities to crash
The objective: minimize the total cost of crashing the activities
Variables: xj : reduction in activity j duration by crashing (j=A,B,..,N)
Constraints:
Total project duration is less than the desired length
The reductions can not be more than the maximum allowed reduction
The reductions must be non-negative
Precedence relationships between activities must hold (we need to define extra variables to write these constraints)
Oct 31, 2012
Wood Saba Vahid

10. Precedence relationships
Define new variables: Yj = start time of activity j (for j=B,C,…,N)
The start time of each activity has to be after all its predecessors are finished, so
Start time of activity j >= (start time + duration) of its immediate predecessor
Duration of each activity j= its normal time – xj
For example, immediate predecessor of activity F is E, so:
YF >= YE + 4 – XE
For activities with more than one predecessor, more inequalities are needed
Oct 31, 2012
Wood Saba Vahid

11. The LP for cost-time trade-off, Example 15
Minimize Z = Normal Cost + (crash cost per week of activity j ∗ reduction in activity j duration)
Subject to:
𝑥 𝐴 ≤1, 𝑥 𝐵 ≤2,… (maximum reduction constraints)
𝑥 𝐴 ≥0, 𝑥 𝐵 ≥0,… (non-negativity constraints)
𝑦 𝐵 ≥0+2− 𝑥 𝐴
𝑦 𝐶 ≥ 𝑦 𝐵 +4− 𝑥 𝐵
𝑦 𝐹𝑖𝑛𝑖𝑠ℎ ≥ 𝑦 𝑀 +2 − 𝑥 𝑀
𝑦 𝐹𝑖𝑛𝑖𝑠ℎ ≥ 𝑦 𝑁 +6 − 𝑥 𝑁 …. (start time constraints/precedence)
𝑦 𝐹𝑖𝑛𝑖𝑠ℎ ≤40 (finish time constraint)
Example 15
Oct 31, 2012
Wood Saba Vahid