Crashing of Networks - Construction Materials & Management - Civil Engineering

Crashing an activity (Crashing the network)

Crashing an activity means reducing the time required to complete that activity by assigning additional resources, with the objective of reducing the overall project completion time. Reducing durations of activities on the critical path may shorten the project, but may also change which path is critical. Crashing increases direct costs; therefore, a trade-off exists between time saved and additional cost incurred.

Key definitions

• Normal time (NT): the expected or standard time normally allowed to complete an activity.
• Normal cost (NC): the cost to complete the activity if it is performed in its normal time.
• Crash time (CT): the minimum feasible duration for the activity when extra resources are applied.
• Crash cost (CC): the direct cost to complete the activity in its crash time.
• Crash cost per time period: (CC - NC) / (NT - CT). This gives the additional cost required to save one unit of time on that activity.

Approaches to crashing a project network

Minimum-time schedule method

1. Determine the critical path using normal times for all activities.
2. Crash every activity from its normal time to its crash time so that each activity is at its minimum duration. This produces the minimum-time schedule (the shortest possible project duration given the individual crash times).
3. If the absolute minimum project time is required but costs need to be reduced, un-crash activities that are not on the critical path beginning with those whose crashing is most expensive (highest crash cost per time) until the desired balance of time and cost is achieved.

Minimum-cost schedule method

1. Determine the critical path using normal times for all activities.
2. Select the activity on the critical path that has the lowest crash cost per time period, and crash that activity as much as feasible (until its crash time), unless another path becomes critical or further crashing is uneconomical.
3. Recompute the critical path(s). Repeat step 2, always choosing the lowest incremental cost per unit time among activities on the then-critical path(s).
4. If more than one path is critical simultaneously, it may be necessary to crash activities on each critical path at the same time. In that case, select the combination of activities that gives the lowest total cost per unit time reduction for the project.

Example

Project planning with uncertain activity times (Three-point estimates and PERT)

When activity durations are uncertain, a three-point estimate is commonly used. For each activity estimate:

• Most optimistic time (a): a duration such that the probability of completion in less than a is very small (about 1%).
• Most likely time (m): the duration considered most likely under normal conditions.
• Most pessimistic time (b): a duration such that the probability of taking longer than b is very small (about 1%).

Using these three estimates, the expected activity time and variance are approximated as follows:

Expected time: t_e = (a + 4m + b) / 6

Variance: var[t_e] = ((b - a) / 6)²

Under the Central Limit Theorem, the sum of activity times on a path can be approximated as normally distributed. Therefore, the project completion time (sum of times on the critical path) has mean equal to the sum of expected times of activities on that path and variance equal to the sum of their variances. This permits calculation of probabilities for meeting target completion dates.

Assumptions

1. Critical path dominance: Durations of activities along the critical path determine the project completion time.
2. Independence and normal approximation: Duration of one activity is assumed independent of others. Using the Central Limit Theorem, the project completion time is assumed approximately normally distributed with mean equal to the sum of expected times on the critical path and variance equal to the sum of the variances of those activities.
3. Multiple critical paths: If a project has more than one critical path, the path with the largest variance should be used when calculating the probability of meeting a particular completion time, because it contributes most to uncertainty.

Note: Considering only the current critical path is a strong assumption. In practice, non-critical paths with large variance in activity durations can become critical due to random variations. Project risk analysis should therefore consider near-critical paths and activities with high variance.

Cost model analysis (Crashing and total project cost)

Crashing increases direct costs by adding resources to shorten activity durations, while indirect costs (overheads, supervision, financing, site supervision, utilities, etc.) typically increase with project duration. The total project cost is the sum of direct and indirect costs. Crashing decisions aim to find the optimum project duration at which total cost is minimum or to determine the least-cost way to achieve a required deadline.

1. Project cost (overview)

Total project cost = Direct cost (labour, materials, equipment, subcontractors) + Indirect cost (site overheads, project management, finance, escalation, lost opportunity). The components and relations between them are often shown graphically.

2. Components of project cost

• Indirect project cost: Expenditures not allocable to individual activities and assessed over the project as a whole. Indirect costs rise as project duration increases; the total indirect cost curve is generally upward sloping with respect to project duration.
• Direct project cost: Costs directly attributable to completing activity work such as labour, materials, plant and equipment. Direct cost normally increases when activities are crashed because of overtime, additional crews, temporary plant, subcontractor premiums, etc.
• Normal time (t_n): Standard activity duration used in the baseline schedule.
• Crash time (t_c): Minimum achievable activity duration by deploying additional resources. Crash time is the lower bound beyond which further resource addition does not shorten the activity.
• Normal cost (C_n): Direct cost to complete the activity in normal time.
• Crash cost (C_c): Direct cost to complete the activity in crash time. For analysis, the direct cost curve between normal and crash times is often approximated as linear (or piecewise linear).
• Cost slope: The incremental cost per unit time saved when crashing an activity. It is used to prioritise which activities to crash.
• Total project cost and optimum duration: The total project cost equals direct cost plus indirect cost. Plotting total cost against project duration typically yields a convex curve with a minimum at the optimum duration. If the project duration is shortened towards the crash limit, direct cost rises and total cost may increase; if duration is extended, indirect costs rise and total cost increases.

Cost Slope = (Crash Cost - Normal Cost) / (Normal Time - Crash Time)

Practical procedure for project crashing (summary of steps)

• Prepare the project network and compute earliest and latest event times using normal durations; identify the critical path(s).
• Compute crash times, normal and crash costs, and the cost slope for each activity where crashing is possible.
• If the objective is minimum time, reduce all activities to crash times (minimum-time schedule). If the objective is minimum additional cost to achieve a given deadline, repeatedly crash the activity (or activities) on the current critical path with the lowest cost slope until the required project duration is reached or no further crashing is feasible.
• After each crashing action, recompute the network (earliest/latest times and critical path(s)), because crashing can change which path is critical.
• When multiple critical paths exist, apply crashing simultaneously to appropriate activities on each critical path so that project duration reduces as intended; select activities to minimise total cost per unit reduction.
• Compare total cost for possible durations and select the optimum duration if objective is total cost minimisation.

Concluding remarks

Crashing is an essential project control technique to meet deadlines or reduce project duration, but it must be applied carefully because of increased direct costs and the risk of shifting criticality among paths. Combining three-point time estimates with variance analysis allows estimation of the probability of meeting revised completion dates. The decision to crash should weigh the marginal cost of time savings (cost slope) against benefits such as reduced indirect costs, contractual penalties avoided, or earlier revenue.