Reduce project duration subject to crash limits and costs practice questions | General Mathematics (Year 12, VCE Units 3 & 4)

Project crashing

Project crashing reduces permitted activity durations to shorten a project. The completion time is set by the longest complete source-to-finish path, so the paths and critical-path set must be recalculated after each reduction

T_{\text{project}}=\max\left(T_1,T_2,\ldots,T_n\right)

Calculate each complete path Add the activity durations on every source-to-finish path $\displaystyle P_1: 8+3+5=16$
Use the longest path The largest path total is the project completion time and identifies a critical path $\displaystyle T_{\text{project}}=\max\left(16,15\right)=16$
Recalculate after a reduction A shortened path can make another path equally long, so recalculate all path totals $\displaystyle \max\left(14,15\right)=15$
Count each activity cost once Multiply each activity reduction by its cost per period and add those activity costs once $\displaystyle 3\times40=120$

Worked example

An activity-on-arrow project has two complete paths. Path S-A-C-T has durations 8 days, 5 days and 6 days. Path S-B-D-T has durations 9 days, 4 days and 5 days. Activity AC is shortened by 2 days at a cost of $45 per day. Find the new project completion time and the added crash cost

\max\left(8+5+6,9+4+5\right)

$\displaystyle 8+5+6=19,\quad 9+4+5=18$Add the durations on each complete path to find the baseline totals
$\displaystyle \max\left(19,18\right)=19$The first path is initially critical because 19 days is greater than 18 days
$\displaystyle \max\left(8+\left(5-2\right)+6,18\right)=\max\left(17,18\right)=18$Shortening AC reduces its path to 17 days, so the second path now sets the completion time at 18 days
$\displaystyle 2\times45=90$The reduction is two days at $45 per day, so the activity cost is $90 once

Common mistakes

Crash the cheapest activity first

✕$\displaystyle \text{shorten the cheapest activity first}$✓$\displaystyle \text{check the current critical paths before choosing an activity}$

A low-cost activity does not shorten the project if it is not on a current critical path

Keep the original critical path fixed

✕$\displaystyle \max\left(17,18\right)=17$✓$\displaystyle \max\left(17,18\right)=18$

The longest path must be recalculated after a reduction because another path can become longest

Count a shared cost for each path

✕$\displaystyle 2\times30\times2=120$✓$\displaystyle 2\times30=60$

The cost belongs to the reduced activity and is counted once, even if that activity lies on more than one path

Difficulty levels

introCrash one named critical activity and recalculate the project completion time

standardChoose between critical activities with different costs per reduced period

challengeHandle multiple critical paths and shared activities while meeting a target duration

examDetermine a minimum-cost reduction vector or diagnose why a proposed reduction has no project effect

Questions about this topic

What is a critical path?+

A critical path is a complete source-to-finish path with the greatest duration. If two or more paths have the same greatest duration, they are all critical

Why must the paths be recalculated after crashing an activity?+

Reducing one path can make another path equal to or longer than it, so the critical path and project completion time can change

How is a crash cost calculated?+

For each reduced activity, multiply the number of periods removed by its cost per period, then add the activity costs once

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, General Mathematics, chapter 14: Flow, matching and scheduling

In this chapter

Read and model a directed flow network 14A Flow networks: Validate a cut and calculate its capacity 14A Flow networks: Determine and certify maximum flow Represent and solve a small bipartite matching problem Find an optimum assignment with the Hungarian algorithm 14B Matching and assignment: Validate a cut and calculate its capacity Translate between precedence data and an activity network Complete a critical path analysis Reduce project duration subject to crash limits and costs 14C Scheduling and critical path analysis: Determine and certify maximum flow