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The Critical Path Method
Quantitative Analysis · Chapter 8: Project Timelines and the Critical PathSection 8.3 · lesson 37 of 5512 min · free with a free account
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In this lesson
Finding the bottleneck, and buying time back at the lowest cost.
- Identify the critical path and why it fixes project duration
- Separate CPM from PERT on durations, focus and output
- Compute a cost slope and run the crashing algorithm
- Locate the minimum-cost schedule on the U-shaped cost curve
Transcript
Everything said in the video, section by section (1,751 words). Read along, or search it.
Project Timelines and the Critical Path · Section 8.3
The Critical Path Method
Some activities in a project can slip a week without anyone noticing. Others cannot slip at all. The ones that cannot form a chain, and that chain decides when the project finishes. It is called the critical path, and finding it is the easy half of this section. The hard half is paying to make it shorter. The Critical Path Method adds money to the network diagram. It lets you trade time for cost, through a procedure called crashing. We will separate C P M from PERT, then build the cost slope from four numbers. Then we run the crashing algorithm, week by week, on a real schedule. By the end you will know exactly what one week of a project is worth.
The Bottleneck
The Longest Path Is the Project
Every project network holds a sequence that determines its total length. It is a chain whose links cannot be shortened by redistributing effort elsewhere. That chain is the critical path. Formally, it is the longest sequence of dependent activities from project start to project finish. Longest. Not most important, not most expensive, not most visible. Two consequences follow, and both are worth memorising. First, delaying any activity on the critical path delays the entire project. Second, compressing that path is the only way to deliver the project sooner. It also carries zero slack, so there is no spare time to absorb a surprise. The method dates to nineteen fifty-seven, built by Morgan R. Walker and James E. Kelley at DuPont. It extends ordinary network analysis by adding the one thing a network diagram lacks: cost.
Two Methods
CPM and PERT on One Network
C P M and PERT are often taught as rivals. They are not. Both came out of late nineteen fifties operations research, and both use the same network notation. What differs is the question each was built to answer. PERT came from the US Navy in nineteen fifty-eight, on the Polaris programme. C P M came a year earlier, from the chemical plants of DuPont. PERT treats durations as probabilistic, using three-point estimates. C P M treats them as deterministic, single values you commit to. So PERT aims at schedule risk, and C P M at the cost-time trade-off. PERT suits research and development, where the work has never been done before. C P M suits repetitive, well-understood work, where durations are reliable but budgets are tight. In modern practice they are rarely used in isolation. PERT computes expected time as a, plus four m, plus b, all over six, and hands those durations to C P M. Microsoft Project and Primavera implement both, seamlessly, under the hood.
Crashing
Two Points, Four Numbers
Crashing is the deliberate shortening of an activity by adding resources. Extra workers, overtime, additional equipment, or subcontractors. Every activity has two reference points on its cost-duration curve. Each point carries a time and a cost, so four numbers describe every activity. Normal time is the duration under normal working conditions: standard crew, single shift, no overtime. Normal cost is the direct cost of performing it in normal time, and it is the lowest that activity can cost. Crash time is the shortest technically feasible duration, using the maximum reasonable resources. Crash cost is the total direct cost at that crash time. It is always higher. Below the crash time no further compression is possible, whatever you are willing to spend. That floor is technical, not financial. Money buys weeks only down to it.
The Cost Slope
The Price of One Week
The most important derived quantity in C P M is the cost slope. It is the additional cost incurred per unit of time saved by crashing the activity. Take crash cost minus normal cost, and divide by normal time minus crash time. Money added on top, divided by weeks bought back. Geometrically it is the slope of the line joining the normal point to the crash point. A low cost slope means the activity is cheap to crash, so consider those first. A high cost slope means it is expensive. Compress those only as a last resort. Now the assumption, and it matters. Standard C P M assumes the cost-duration relationship is linear between the two points. In reality the curve is often convex: each successive week saved is dearer than the last. Linearity is still a reasonable working approximation, and it is the textbook convention. It is what examinations and most project management software use. Just know it is an assumption.
The Algorithm
Crash, Re-check, Repeat
Crashing is not a one-step exercise. The loop is the whole point. Step one. Identify all critical paths, using normal durations. There may be more than one path of equal length, so check for ties. Step two. Among activities on the critical paths, find the smallest cost slope. If several critical paths exist, the candidate must lie on all of them. Otherwise you select one candidate from each path, and pay for both. Step three. Crash that activity by one time unit, or down to its crash limit. Add the marginal cost to the cumulative crash cost. Step four. Recompute the paths, before you do anything else. Crashing may shorten the original critical path enough that a different path is now longest. Miss that, and you keep spending on a route that stopped being the bottleneck. Step five. Check the stopping condition, then go back to step one.
The Economics
The Minimum-Cost Schedule
Why not simply crash everything to its limit? Because crashing is not free. But neither is time. Every week saved by crashing costs money in direct cost. Every week saved also removes a week of indirect cost. Indirect cost is overhead, site rental, supervision, equipment hire, financing, and the opportunity cost of capital. So as the project is compressed, direct costs rise and indirect costs fall. Add the two together, and the total cost curve forms a characteristic U-shape. The minimum of that U is the minimum-cost schedule, the economically optimal duration. The rule is one line. Stop crashing when marginal crash cost equals marginal indirect-cost saving. Push past that point, and every extra week purchased is a net financial loss. The schedule keeps shortening. The money stops making sense. One caveat. Contracts, penalty clauses and reputation can justify crashing well beyond that point.
Worked Example
6 Activities, 20 Weeks, a 16-Week Deadline
Now the full example, and it is worth working through by hand. A project has a normal duration of twenty weeks. The client requires sixteen. Six activities, A through F, each with a normal and a crash figure for time and cost. Take activity A. Five weeks normally, three weeks crashed. Its cost rises from one thousand pounds to eighteen hundred. Eight hundred pounds extra, for two weeks bought, is four hundred pounds a week. Activity C is the interesting one. Seven weeks normal, four weeks crashed. Fifteen hundred pounds normal, twenty-four hundred crashed. Nine hundred pounds over three weeks is three hundred pounds per week. That is the cheapest slope in the project, and C can surrender three whole weeks. Compute every slope before you crash anything. The ranking is the entire strategy.
The Network
Two Paths, One Critical
Next, draw the network from the predecessor list. A and B both start immediately, so the network opens with two branches. C follows A. D follows B. E waits for both C and D. F follows E, so the branches merge at E and finish together. That gives two routes from start to end. Path one is A, C, E, F: five plus seven plus four plus four, which is twenty weeks. Path two is B, D, E, F: three plus four plus four plus four, which is fifteen. Path one is critical. Path two carries five weeks of slack. So the twenty-week duration is set entirely by path one. Crashing B or D would compress a route with five weeks already to spare. Notice too that E and F sit on both paths, so crashing either one shortens both.
Iterative Crashing
Four Weeks, Bought One at a Time
We need to remove four weeks from a twenty-week schedule. The algorithm does it one week at a time, re-checking after each. Iteration one. The critical path is A, C, E, F. The cheapest activity on it is C, at three hundred pounds a week. Crash C by one week. The project drops to nineteen weeks, for three hundred pounds. Iterations two and three repeat it. Eighteen weeks, then seventeen, at six hundred and then nine hundred pounds cumulative. C has now gone from seven weeks down to four, which is its crash limit. Path two is still fifteen weeks, so A, C, E, F remains the critical path. For the final week, A and E both offer a cost slope of four hundred pounds. A common tie-break is to crash the activity that touches the fewest other paths. Here the tie is genuinely cost-neutral: crashing A instead of E also reaches sixteen weeks for the same money. We take E. Sixteen weeks, thirteen hundred pounds of crashing in total.
Final Result
16 Weeks for £1,300 More
So where does that leave the project? Original duration, twenty weeks. Crashed duration, sixteen. Four weeks of compression. The normal direct costs sum to six thousand five hundred pounds. The crashing adds thirteen hundred pounds on top of that. Total direct cost at the sixteen-week schedule is seven thousand eight hundred pounds. Now divide the thirteen hundred by the four weeks it bought. The effective cost is three hundred and twenty-five pounds per week saved. That single number is the one you take to the client. If indirect cost saved per week beats three hundred and twenty-five pounds, the crash pays for itself. Overhead, supervision and financing all count towards that saving. If the saving is smaller, crashing is a loss on pure cost grounds. Then the case has to rest on contract or strategy: a penalty clause avoided, or market share captured by delivering early.
Key Takeaways
What to Carry Forward
Six things to carry out of this section. The critical path is the longest sequence of dependent activities, and it holds zero slack. It determines the project duration, and nothing else does. C P M uses deterministic durations and focuses on the cost-time trade-off. PERT uses probabilistic durations and focuses on schedule risk. In practice, combine them. Crashing shortens an activity by deploying extra resources. The cost slope is crash cost minus normal cost, over normal time minus crash time. The algorithm always crashes the lowest-slope activity on the current critical path. The word current is doing real work there, because the critical path can change every step. The minimum-cost schedule sits at the bottom of the U-shaped total cost curve. In the example, twenty weeks became sixteen for thirteen hundred pounds. Three weeks of C at three hundred, plus one week of E at four hundred.
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