Question 6 of 7: Machinery-Upgrade Downtime — CPM Network
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Technical Examinations — December 2019 — 17-Ind-A4 Production Management. Three-hour, closed-book exam; Casio or Sharp approved calculators only. Format: seven questions, each worth 20 marks (sub-part weights per the front-page marking scheme); candidates do two questions from Section A and three from Section B, and only the first five questions appearing in the answer book are marked. All seven are solved below for completeness. The paper asks for point-form answers wherever possible; the solutions below use full working for clarity.
Reference texts: Liker, The Toyota Way, and Shingo, A Revolution in Manufacturing: The SMED System — JIT, 5S/andon/poka-yoke/SMED/TPM and lean root-cause analysis; Niebel & Freivalds, Methods, Standards, and Work Design — process charting and methods analysis; Nahmias & Olsen, Production and Operations Analysis (7th ed., Waveland/McGraw-Hill) — forecasting, lot sizing (Wagner–Whitin) and aggregate planning; Hillier & Lieberman, Introduction to Operations Research (11th ed.) — project scheduling (CPM/PERT); Pinedo, Scheduling: Theory, Algorithms, and Systems (5th ed.) — parallel-machine scheduling and days-off workforce scheduling.
Given. An 11-activity precedence network. Activity A (order/receive the new machines, 480 h) precedes B but does not itself require the line to be stopped — ordering and delivery run alongside normal production. Production down-time begins at B ("Stop production") and ends when L ("Start production") completes.
Activity
Description
Precedes
Duration
A
Order & receive new machines P, Q
B
480 h
B
Stop production
C, G, H
2 h
C
Remove old machine Q
D
2 h
D
Prepare machine base for Q
E
8 h
E
Install new machine Q
F
2 h
F
Test & commission machine Q
L
3 h
G
Remove old machine P
H
1 h
H
Prepare new electrical feed for P
J, L
5 h
J
Install new machine P
K
2 h
K
Commission & test machine P
L
12 h
L
Start production
—
1 h
Find. (a) The total production down-time required; (b) a revised precedence that minimizes down-time, justified.
Figure 3 — Activity network. A (procurement) precedes B but runs before the stoppage; down-time is measured from B to L. Edge labels are the FROM-activity's duration (hours).
Approach. Run a forward pass through the down-time sub-network (activities B–L only — A's 480 h happens before the line stops, so it is excluded from the down-time clock) to find the critical path and total down-time; then re-run the forward pass on a revised precedence that removes an unnecessary serial dependency from that critical path.
Forward pass, original precedence (part a). Setting $t=0$ at B (production stops):
$$EF_B=2,\ EF_C=4,\ EF_D=12,\ EF_E=14,\ EF_F=17,\ EF_G=3,\ EF_H=8,\ EF_J=10,\ EF_K=22,$$
$$EF_L=\max(EF_F,\,EF_K,\,EF_H)+1=\max(17,22,8)+1=\boxed{23\ \text{hours}}.$$
The critical (zero-slack) path is $B\to G\to H\to J\to K\to L$ (Machine-P chain: $2+1+5+2+12+1=23$ h), strictly longer than the Machine-Q chain $B\to C\to D\to E\to F\to L$ ($2+2+8+2+3+1=18$ h, 5 h of slack).
Down-time answer (part a).
$$\boxed{\text{Production down-time}=23\ \text{hours}}$$
(Activity A's 480 h of procurement lead time is not counted — it runs concurrently with normal production, before B stops the line.)
Revise the precedence (part b). Activity H ("prepare new electrical feed for P," 5 h) is the largest single item on the critical Machine-P chain besides K, and inspection shows it does not actually require the old machine P to already be removed — wiring a new electrical circuit/panel for the replacement machine's location can physically be done entirely before the line stops, exactly like procurement activity A (which already has 480 h of lead time to absorb it), since the wiring work does not touch the currently-running machine P itself. Machine P's physical removal (G) must still happen after the stoppage, and installing the new machine (J) still needs both the old machine gone (G) and the new feed ready — but since H is now finished before the down-time clock even starts, it drops out of the down-time network altogether (the same treatment as A) and J only waits on G.
Forward pass, revised precedence. With H removed from the down-time clock (completed in advance) and J's only remaining down-time predecessor G:
$$EF_G=3,\ EF_J=EF_G+2=5,\ EF_K=EF_J+12=17,$$
$$EF_L=\max(EF_F,\,EF_K)+1=\max(17,17)+1=\boxed{18\ \text{hours}}.$$
Result. Down-time falls from 23 to 18 hours, a saving of $23-18=\boxed{5\ \text{hours}\ (22\%)}$. The Machine-Q chain (B-C-D-E-F-L $=18$ h) and the revised Machine-P chain (B-G-J-K-L $=2+1+2+12+1=18$ h) now tie exactly at the new critical value, so both chains are simultaneously critical; no further reduction is available without shortening K (12 h, commissioning and testing the new machine P) or the Machine-Q chain's D (8 h, preparing the machine base), both of which are genuine hands-on work that must happen after the corresponding old machine is removed and cannot be pre-staged during the procurement lead time.
Scenario
Critical path(s)
Down-time
Original precedence
B–G–H–J–K–L
23 h
Revised (H prepped fully in advance, like A)
B–C–D–E–F–L and B–G–J–K–L (tied)
18 h
Saving
5 h (22%)
Check: the revision assumes the new electrical feed can be routed/installed at a location or panel independent of machine P's current footprint, so wiring work does not require P to be shut down or removed first — a reasonable assumption for most plant electrical work, but one that should be confirmed against the specific floor layout before committing to the revised schedule.
