Linear Relationship Between Time and Cost .. project'--from initial planning through construction execu- tion., Time .. Total project time can be shortened by "crashing" Collins, F. Thomas, Manual Critical Path Techniques for. relationships. •. Could also Critical path determines the minimum completion time for a project. •. Use forward .. Project Crashing and Time-Cost. Analysis. Project Crashing and Time-Cost Trade-Off If we assume that the relationship between crash cost and crash time is linear, then activity paths as we reduce individual activities, a condition that makes manual crashing very cumbersome.
When two paths simultaneously become critical, activities on both must be reduced by the same amount. If we reduce the activity time beyond the point where another path becomes critical, we may be incurring an unnecessary cost. This last stipulation means that we must keep up with all the network paths as we reduce individual activities, a condition that makes manual crashing very cumbersome.
For that reason we will rely on the computer for project crashing; however, for the moment we pursue this example in order to demonstrate the logic of project crashing. It turns out that activity can be crashed by the total amount of 5 weeks without another path becoming critical, since activity is included in all four paths in the network.
The revised network is shown in the following figure. Since we have not reached our crashing goal of 30 weeks, we must continue and the process is repeated. Activity can be crashed a total of 3 weeks, but since the contractor desires to crash the network only to 30 weeks, we need to crash activity by only 1 week. Crashing activity by 1 week does not result in any other path becoming critical, so we can safely make this reduction. Crashing activity to 7 weeks i.
Suppose we wanted to continue to crash this network, reducing the project duration down to the minimum time possible; that is, crashing the network the maximum amount possible. We can determine how much the network can be crashed by crashing each activity the maximum amount possible and then determining the critical path of this completely crashed network.
For example, activity is 7 weeks, activity is 5 weeks, is 3 weeks, and so on.
Network analysis - cost/time tradeoff
The critical path of this totally crashed network is with a project duration of 24 weeks. This is the least amount of time the project can be completed in. It is basically a trial-and-error approach useful for demonstrating the logic of crashing. It quickly becomes unmanageable for larger networks. This approach would have become difficult if we had pursued even the house building example to a crash time greater than 30 weeks, with more than one path becoming critical.
When more than one path becomes critical, all critical paths must be reduced by an equal amount. Since the possibility exists that an additional path might become critical each time the network is reduced by even one unit of time e. The General Relationship of Time and Cost In our discussion of project crashing, we demonstrated how the project critical path time could be reduced by increasing expenditures for labor and other direct resources.
The objective of crashing was to reduce the scheduled completion time to reap the results of the project sooner. However, there may be other reasons for reducing project time.
As projects continue over time, they consume indirect costs, including the cost of facilities, equipment, and machinery, interest on investment, utilities, labor, personnel costs, and the loss of skills and labor from members of the project team who are not working at their regular jobs.
There also may be direct financial penalties for not completing a project on time. For example, many construction contracts and government contracts have penalty clauses for exceeding the project completion date.
In general, project crashing costs and indirect costs have an inverse relationship; crashing costs are highest when the project is shortened, whereas indirect costs increase as the project duration increases. The advantage of using linear programming to crash a project is that we can automatically guarantee that, for any particular project completion time, we have achieved that time by crashing in a minimum cost fashion.
Crashing using incremental costs may, because of the difficulty of dealing with multiple critical paths, not lead us to a minimum cost solution for each possible project completion time. The package output giving the cost associated with crashing the project from its normal completion time of 24 weeks to 19 weeks for example is given below.
It can be seen that the minimum cost way to achieve an overall project completion time of 19 weeks is by crashing activity 5 by one week, activity 8 by three weeks and activity 9 by one week. The output below shows the minimum cost way of achieving the lowest possible overall project completion time of 16 weeks.
It can be seen that this can be done for a cost of This contrasts with the cost of associated with using all activities at their crash times. The difference arises because it is not necessary to crash all activities to their maximum extent to achieve an overall project completion time of 16 in this case activity 2 does not need to be crashed. By varying the number of weeks by which we crash the project we can construct the graph shown below.
In that graph we have plotted, for each possible project completion time, the minimum cost associated with achieving that completion time. Note here that this graph contains three distinct straight line segments 16 to 18, 18 to 21, 21 to This arises because of the linear relationship that was assumed to hold between cost and completion time for each activity.
Note there that the package merely provides information, in this case the cost of the project for all possible completion times between 16 and 24 weeks. It does not tell you which completion time you should choose as you can have any completion time between 16 and 24 weeks provided you are prepared to pay for it! What the package does is enable you, as the project manager, to make an informed choice about the completion time to have. Activity splitting For the network considered above we have seen that the minimum possible completion time associated with the maximum cost is 16 weeks.
But what if we really wanted a completion time of 15 weeks - is there any possible way of achieving that? The simple answer is NO, but with a caveat, not with the project as currently represented by the network. It may be that we can change our project network, opening up the possibility of potentially reducing the overall project completion time below 16 weeks.
A common approach to do this is activity splitting. In activity splitting we typically examine each critical activity since critical activities are the determining factor in overall project completion and see if they can be split into two or more separate activities. For example, for the network considered before we might decide that activity 1, requiring 6 weeks can be split as below: Here we have split this activity into 4 sub-activities 1a,1b,1c,1d and their precedence relationships.
It can be seen that by so doing we have actually increased the time required to complete activity 1 to 7 weeks, from the original 6 weeks. However, it may be that this subdivision of activity 1, when considered from the cost crashing point of view, gives us more flexibility.
For example before subdivision we could only crash activity 1 down from 6 weeks to 4 weeks. If we could now crash activity 1b by 2 weeks and activity 1c by 3 weeks then we could crash activity 1 down from 7 weeks to 3 weeks - hence potentially reducing the overall project duration below the 16 week barrier we previously encountered. Obviously the practicality of subdividing, and then crashing, critical activities depends upon the context but the above example does illustrate that sometimes close examination of critical activities with a view to subdivision can pay benefits.
Exploring the package Because cost crashing can be modelled and solved via linear programming we have a number of alternative options available, as the package illustrates. For example we might be interested in finding the minimum time in which we can complete the project subject to a constraint limitation upon the total cost. This is illustrated below using the package for a total cost of It can be seen above that the minimum completion time subject to a cost constraint of is The package also allows us to balance rewards in meeting a target desired project completion time as against penalties for missing this completion time.
The example below illustrates this where the desired project completion time is 20 weeks, each week we are late complete after week 20 costs us 50 and each week we are early complete before week 20 earns us a reward of It can be seen that the optimum least cost completion time in this situation is 18 weeks, leading to a total minimum cost of For example, let us return to the original network we had before with the original crashing data as illustrated below.
Chapter 17, Head 5
Deciding to complete the project in 18 weeks gives the output below. So, for example, activity 1, with a suggested time of 6 costs usi. Activity 2, with a suggested time of 2 costs usi. We have stressed above the weeks during which an activity is done. Whilst any activities which are critical must be performed at precise times in order to complete the overall project in 18 weeks is the same true of non-critical activities? For this project with a completion time of 18 weeks the Activity Criticality Analysis is Activity 2, for example which requires 2 weeks is non-critical and has a slack of 6 weeks.
It can potentially be started at any time between week 0 its earliest start time and week 6 its latest start time without affecting the overall project completion time.
If we were to start it in week 4, for example, the cost of 50 per week associated with activity 2 would not be incurred until week 4.
This may potentially be of benefit why incur a cost before you need to? In that figure the ES line is shown in purple, and the LS line in blue. The gap between the cumulative ES and LS lines represents flexibility - cost can be adjusted within the ES and LS limits by artificially delaying the start of non-critical activities. It is important to note however that artificially delaying the start of non-critical activities and hence incurring activity costs later is not free.
Rather it costs us.