Question 3 of 7: Assembly Line Balancing (RPW Technique), Balance Delay, and Balancing-Technique Modifications

Question 3: Assembly Line Balancing (RPW Technique), Balance Delay, and Balancing-Technique Modifications (20 marks: i–10, ii–5, iii–5)

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.

(i) Line Balancing by the Ranked Positional Weight (RPW) Technique

Given. 12 task elements with times and immediate-predecessor precedence as tabulated above (total content time $\sum t_i=70$ min); the line is operated 7 h/day, 5 days/week, for a required output of 130 units/week.

Find. The theoretical minimum and actual number of stations, the station assignment (schematic), and the resulting line efficiency.

Approach. Convert the weekly output requirement to a cycle time from the available line-operating time, find the theoretical minimum station count, rank every element by its Ranked Positional Weight, assign elements to stations in RPW-rank order — adding the highest-ranked precedence-ready element that still fits the station's remaining cycle time — then compute the resulting line efficiency.

1. Cycle time from the required weekly output. The line runs 7 h/day × 5 days/week $=35$ h/week $=2100$ min/week (the company's 8-hour working day is not the line's OPERATING time — only the stated 7 h/day the line itself runs is relevant). $C=\dfrac{2100\ \text{min/week}}{130\ \text{units/week}}=\boxed{16.154\ \text{min/unit}}$.
2. Theoretical minimum number of stations. $N_{min}=\left\lceil\dfrac{\sum t_i}{C}\right\rceil=\left\lceil\dfrac{70}{16.154}\right\rceil=\lceil 4.33\rceil=\boxed{5\ \text{stations}}$.
3. Ranked Positional Weight of every element. $RPW_i=t_i+\sum_{j\in\text{followers}(i)}t_j$ (own time plus every element, direct or indirect, that must come after it):

   Element	1	2	3	6	4	7	5	8	9	10	11	12
   RPW (min)	70	58	31	29	27	25	20	18	18	17	13	7

   Elements 8 and 9 tie at RPW $=18$; the tie is broken by precedence readiness during assignment (Step 4), not by the ranking itself.
4. Station assignment (largest-candidate rule). Working down the RPW list, add each precedence-ready element to the open station if it fits the remaining cycle time ($C=16.154$ min); otherwise leave it for a later station and try the next-ranked ready element:

   Station	Elements (assignment order)	Station time (min)	Idle (min)
   1	1	12	4.154
   2	2, 3, 4, 5	16	0.154
   3	6, 9	13	3.154
   4	7, 8, 10	16	0.154
   5	11, 12	13	3.154

   Every station time is $\le C=16.154$ min and every precedence requirement is respected (e.g. element 7 needs 3 and 4, both assigned to station 2; element 10 needs 9 and 6, both assigned to station 3; element 11 needs 8 and 10, both assigned to station 4). This uses exactly $N_{actual}=\boxed{5\ \text{stations}}$ — the theoretical minimum from Step 2 is achieved.
5. Line efficiency. $$\text{Efficiency}=\frac{\sum t_i}{N_{actual}\cdot C}=\frac{70}{5(16.154)}=\frac{70}{80.77}=\boxed{86.7\%}$$ Balance delay $=100\%-86.7\%=\boxed{13.3\%}$. The line's own achieved cycle (its slowest station, 16 min) is under the allowed $C=16.154$ min, so the design comfortably meets the 130-units/week target — at a 16 min actual station cycle the line delivers $2100/16=131.3$ units/week, slightly above the 130 required.

(ii) Reasons for the Increase in Line Balancing Delay in Progressive Assembly

Balance (balancing) delay is the fraction of total station-time capacity that goes idle because a fixed cycle time cannot be filled perfectly evenly across every station (the trade-off noted in Question 2(iii)). In progressive (continuously paced) assembly, several factors push this delay up: (1) Increasing the number of stations for a fixed total work content shrinks the average station's capacity, so the same, largely indivisible task-element times occupy a growing share of each station's cycle — a small element that was a modest fraction of a large station becomes a large, hard-to-absorb fraction of a small one, and delay rises even though the line is, in principle, divided more finely. (2) A required cycle time that does not divide evenly into the individual element times or the total content time forces stations to carry unavoidable slack, since an element cannot be split between two stations (Station 1 above, at 12 of 16.154 min, illustrates this — no other ready element fits the remaining 4.15 min). (3) Tight precedence relationships and zoning restrictions (elements that must or must not share a station) reduce the number of feasible groupings available to the balancing technique, so fewer combinations exist that come close to filling every station to capacity. (4) Variability in individual element times (operator pace differences, machine-paced sub-operations, material variation) means a station sized for the average element time is sometimes forced to run under capacity to avoid exceeding the cycle time on a slower cycle. Each of these narrows the achievable efficiency below 100%, and RPW (or any balancing technique) can only minimize, not eliminate, the resulting delay.

(iii) Modifications to the Standard Technique to Balance Assembly/Flow Lines

Where a straight RPW/largest-candidate assignment still leaves significant balance delay, several modifications to the standard technique reduce it further, each relaxing one of the assumptions behind the basic method: (1) Splitting large, methods-engineering-divisible task elements into two or more smaller sub-elements (via tooling, fixture, or method redesign) so they can be distributed more flexibly across stations, directly attacking the indivisible-element cause of delay. (2) Combining or sharing operators between adjacent under-loaded stations — a utility/floater operator cross-trained on neighbouring stations absorbs idle capacity that a rigid one-operator-per-station assignment cannot. (3) Duplicating (paralleling) a station whose element(s) cannot be reduced below the cycle time, running two identical stations in parallel each producing at half the rate, which relieves the bottleneck without forcing every other station down to that element's time. (4) Adopting task-sharing / self-balancing arrangements such as a bucket-brigade or U-shaped line, where an operator who finishes early moves forward to help the next station rather than sitting idle, letting the line balance itself dynamically rather than through one fixed station assignment. (5) Relaxing the cycle time slightly above the strict minimum implied by the output requirement, trading a small loss of rated output for a cycle time that divides more evenly into the element times and materially reduces delay. Each of these is a genuine engineering change (to method, staffing, or layout), not a re-run of the same RPW ranking — which is why they are described as modifications to, rather than applications of, the standard technique.