Project Scheduling and Analysis: 24-MMP-B8 Rock Slope Engineering, Undated paper, Q4

Question 4: Project Scheduling and Analysis (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Task list (verified against the printed paper)
Task	Description	Duration (mo)	Depends on
1	Drive ramp to base of new zone	11	none
2	Develop/equip new raises for hoisting and ventilation	12	1
3	Develop new u/g exploration drilling gallery	2	1
4	Complete new u/g exploration drilling program	8	3
5	Develop ore body model and mining schedule	2	4
6	Reconfigure mine ventilation system for new zone	4	2
7	Expand u/g diesel powered equipment fleet	4	6
8	Develop upper mining level for new zone	8	7
9	Develop lower mining level for new zone	12	7
10	Develop slot raises for initial stope blocks	2	8, 9
11	Drill open stoping blastholes for initial 2 stopes	1	10, 5
12	First production from stopes in new zone	0 (milestone)	11, 2

Find. (a) a Gantt chart of all 12 tasks; (b) the critical path and shortest project duration by CPM.

Approach. Compute each task's early start/finish by a forward pass (ES = latest EF of its predecessors, EF = ES + duration), read the project duration off the final milestone's EF, then run a backward pass (LF = earliest LS of successors, LS = LF − duration) to find zero-float tasks, which form the critical path.

Fig. 4.1 — Gantt chart of all 12 tasks (bar position = early start, bar length = duration). Red bars carry zero float and form the critical path.

Forward pass (early start/finish). Working task-by-task in dependency order: $$ES_1=0,\ EF_1=11 \quad ES_2=11,\ EF_2=23 \quad ES_3=11,\ EF_3=13 \quad ES_4=13,\ EF_4=21 \quad ES_5=21,\ EF_5=23$$ $$ES_6=23,\ EF_6=27 \quad ES_7=27,\ EF_7=31 \quad ES_8=31,\ EF_8=39 \quad ES_9=31,\ EF_9=43$$ $$ES_{10}=\max(EF_8,EF_9)=43,\ EF_{10}=45 \qquad ES_{11}=\max(EF_{10},EF_5)=\max(45,23)=45,\ EF_{11}=46$$ $$ES_{12}=\max(EF_{11},EF_2)=\max(46,23)=46,\ EF_{12}=46$$ $$\boxed{\text{Shortest project duration} = EF_{12} = 46\ \text{months}}$$
Backward pass (late start/finish) and critical-path identification. Setting $LF_{12}=46$ and working back through each task's successors, the tasks with zero float ($ES=LS$) are $$\boxed{\text{Critical path: } 1\rightarrow2\rightarrow6\rightarrow7\rightarrow9\rightarrow10\rightarrow11\rightarrow12,\qquad 11+12+4+4+12+2+1+0=46\ \text{months}}$$ Tasks 3, 4, 5 and 8 carry positive float (e.g. Task 8's chain 7→8→10 finishes at month 39/41, four months ahead of Task 9's parallel chain that actually drives Task 10's start) and are not on the critical path.

Fig. 4.2 — CPM activity-on-node network with early/late start and finish per task; red boxes and arrows mark the zero-float critical path.

Question 4 — final results
Item	Result
Critical path	1→2→6→7→9→10→11→12
Shortest project duration	46 months
Non-critical tasks (positive float)	3, 4, 5, 8

Topic. Gantt charting and the Critical Path Method (CPM) for a linear, dependency-driven project schedule.

Key relations. Forward pass: $ES=\max(EF\ \text{of predecessors})$, $EF=ES+duration$. Backward pass: $LF=\min(LS\ \text{of successors})$, $LS=LF-duration$. Float $=LS-ES$; the critical path is the chain of zero-float tasks and equals the project's minimum duration.

Why this works. Two parallel task chains (here, the exploration/ore-body-model chain 1→3→4→5 vs. the ventilation/fleet/mining-level chain 1→2→6→7→9) both feed the same downstream task (10); the schedule can only move as fast as its slowest feeding chain, so the longer chain (here 1-2-6-7-9, finishing at month 43 vs. 1-3-4-5's month 23) sets the pace and absorbs all of that task's float.

Common pitfall. Forgetting that a task with two predecessors (like Task 10, fed by both Task 8 and Task 9) can only start once BOTH finish (use the max, not either individually); and assuming the task list's numbering order is the same as its dependency/schedule order.

Source. Project Management Institute, A Guide to the Project Management Body of Knowledge (PMBOK Guide) (Critical Path Method); Hartman & Mutmansky, SME Mining Engineering Handbook (mine project scheduling).

Question 4 of 5: Project Scheduling and Analysis

Question 4: Project Scheduling and Analysis (20 marks)